#7815867

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

4839 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);(1,2,5,6)(3,7,4,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,24)(20,25)(21,27)(22,23)(26,31)(28,30)(29,32);(1,3,5,4)(2,7,6,8)(10,14)(12,15)(18,23)(21,31)(24,32)(25,28);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,24)(20,26)(21,28)(22,29)(23,32)(25,31)(27,30);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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, 7723063);