#7657997

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	#36576	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4798 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(18,24)(20,27)(25,28)(29,31);(2,3)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,26)(21,25)(22,27)(23,24)(29,30)(31,32);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,19,23,21)(18,25,24,28)(20,29,27,31)(22,30,26,32);(1,4,5,8)(2,6,3,7)(17,20,23,27)(18,26,24,22)(19,29,21,31)(25,32,28,30);(17,21,23,19)(18,25,24,28)(20,31,27,29)(22,30,26,32);(17,22,23,26)(18,27,24,20)(19,30,21,32)(25,31,28,29)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=i,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565193);