#7622683

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	C₂×#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,25)(17,26)(18,27)(22,33)(23,31)(24,32)(28,35)(29,36)(30,37)(34,39)(38,40);(3,8)(5,7);(2,6)(5,7)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239381, #72239463, #72239467, …
maximal normal subgroup of	#72239381, #72239463, #72239467, …
maximal quotient of	#72343785

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529879);