#7711232

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4884 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7618428);