#7631692

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	#36318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=h,f^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, 7538888);