#7630830

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	#34624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4998 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538026);