#7622578

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	#3449	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1940 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)(5,6,9,15,10,16,24,36,11,14,25,34,26,35,46,56)(7,12,19,29,20,30,42,52,18,28,40,50,41,51,60,65)(8,13,22,32,23,33,45,55,21,31,43,53,44,54,61,66)(17,27,37,48,38,49,57,64,39,47,58,62,59,63,67,68);(2,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,27)(14,28)(15,29)(16,30)(21,39)(22,37)(23,38)(24,42)(25,40)(26,41)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(43,58)(44,59)(45,57)(46,60)(53,62)(54,63)(55,64)(56,65)(61,67)(66,68);(5,8,9,22,10,23,24,45,11,21,25,43,26,44,46,61)(6,13,15,32,16,33,36,55,14,31,34,53,35,54,56,66)(7,17,19,37,20,38,42,57,18,39,40,58,41,59,60,67)(12,27,29,48,30,49,52,64,28,47,50,62,51,63,65,68);(7,18)(8,21)(12,28)(13,31)(19,40)(20,41)(22,43)(23,44)(29,50)(30,51)(32,53)(33,54)(42,60)(45,61)(52,65)(55,66);(6,14)(12,28)(13,31)(15,34)(16,35)(27,47)(29,50)(30,51)(32,53)(33,54)(36,56)(48,62)(49,63)(52,65)(55,66)(64,68)

Polycyclic presentation:

2^9|a^2=g,c^2=g,g^2=h,h^2=i,b^a=bf,e^a=ei,d^b=di,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343591, #72343825, #72343827
maximal normal subgroup of	#72343591, #72343825, #72343827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529774);