#7679230

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	#45303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4600 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7586426);