#7631087

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	#36374	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5108 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)(3,8,4,7)(9,10,13,12)(11,15,16,14)(17,18,22,23)(19,24,26,20)(21,29,30,25)(27,28,32,31);(1,3,6,4)(2,7,5,8)(9,11,13,16)(10,14,12,15)(18,23)(19,26)(25,29)(27,32);(1,4,6,3)(2,8,5,7)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32);(1,2,6,5)(3,7,4,8)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,32)(30,31);(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,15,16,14)(17,21,22,30)(18,25,23,29)(19,27,26,32)(20,28,24,31)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538283);