#7826307

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

4860 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

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, 7733503);