#7622689

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×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319161, #72320147, #72320157
maximal normal subgroup of	#72319161, #72320147, #72320157

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529885);