#7737837

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	#59696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4789 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)(9,10)(11,12)(13,14)(15,16)(17,18)(19,26)(20,21)(22,24)(23,25)(27,32)(28,29)(30,31);(1,2,5,6)(3,7,4,8)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32);(10,14)(11,15)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32);(1,3,5,4)(2,7,6,8)(9,11,13,15)(10,12,14,16)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,14)(13,16)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32);(1,4,5,3)(2,8,6,7)(17,22,23,31)(18,24,25,30)(19,27,20,28)(21,29,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7645033);