#7622823

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622820
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

(1,2)(3,4)(5,6,12,18)(7,19,23,47)(8,14,24,38)(9,17,11,15)(10,29,32,42)(13,34,37,22)(16,28,44,31)(20,46,48,66)(21,51,33,43)(25,41,27,39)(26,54,52,63)(30,45,58,36)(35,59,60,50)(40,53,62,56)(49,67,57,64)(55,65,68,61);(5,7)(6,13)(8,20)(9,21)(10,30)(11,33)(12,23)(14,35)(15,36)(16,43)(17,45)(18,37)(19,29)(22,31)(24,48)(25,49)(26,55)(27,57)(28,34)(32,58)(38,60)(39,61)(40,64)(41,65)(42,47)(44,51)(46,54)(50,56)(52,68)(53,59)(62,67)(63,66);(2,4)(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,24)(13,35)(15,39)(16,40)(17,41)(18,38)(19,46)(21,49)(22,50)(23,48)(28,53)(29,54)(30,55)(31,56)(32,52)(33,57)(34,59)(36,61)(37,60)(42,63)(43,64)(44,62)(45,65)(47,66)(51,67)(58,68);(8,24)(14,38)(20,48)(25,27)(26,52)(35,60)(39,41)(40,62)(46,66)(49,57)(50,59)(53,56)(54,63)(55,68)(61,65)(64,67);(5,9,12,11)(6,15,18,17)(7,21,23,33)(8,25,24,27)(10,28,32,31)(13,36,37,45)(14,39,38,41)(16,42,44,29)(19,43,47,51)(20,49,48,57)(22,30,34,58)(26,53,52,56)(35,61,60,65)(40,63,62,54)(46,64,66,67)(50,55,59,68)

Polycyclic presentation:

2^9|a^2=i,e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,h^a=hi,f^b=fh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343353, #72343359, #72343374, …
maximal normal subgroup of	#72343353, #72343359, #72343374, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530019);