#7622774

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	#3460	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1305 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)(9,10,14,20,15,13,28,29,16,21,31,37,22,30,19,36)(11,17,24,33,12,18,27,35,23,26,38,39,25,34,32,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,15)(13,18)(14,24)(16,23)(19,32)(20,33)(21,26)(22,25)(27,28)(29,35)(30,34)(31,38)(36,40)(37,39);(3,8)(5,7)(9,12)(10,18)(11,22)(13,26)(14,27)(15,23)(16,25)(17,30)(19,24)(20,35)(21,34)(28,38)(29,39)(31,32)(33,36)(37,40);(2,6)(5,7)(11,23)(12,25)(17,26)(18,34)(24,38)(27,32)(33,39)(35,40);(9,13,14,29,15,21,28,37,16,30,31,36,22,10,19,20)(11,18,24,35,12,26,27,39,23,34,38,40,25,17,32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529970);