#7631706

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#34964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538902);