#7622579

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

1936 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)(7,12,19,28,20,29,42,50,18,30,40,51,41,52,60,65)(8,13,22,32,23,33,45,55,21,31,43,53,44,54,61,66)(17,27,38,48,39,49,59,64,37,47,57,62,58,63,67,68);(2,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,27)(14,30)(15,28)(16,29)(21,37)(22,38)(23,39)(24,42)(25,40)(26,41)(31,47)(32,48)(33,49)(34,51)(35,52)(36,50)(43,57)(44,58)(45,59)(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)(7,17,19,38,20,39,42,59,18,37,40,57,41,58,60,67)(12,27,28,48,29,49,50,64,30,47,51,62,52,63,65,68);(8,21)(13,31)(17,37)(22,43)(23,44)(27,47)(32,53)(33,54)(38,57)(39,58)(45,61)(48,62)(49,63)(55,66)(59,67)(64,68);(6,14)(7,18)(13,31)(15,34)(16,35)(17,37)(19,40)(20,41)(32,53)(33,54)(36,56)(38,57)(39,58)(42,60)(55,66)(59,67)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343591, #72343839
maximal normal subgroup of	#72343591, #72343839

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529775);