#7655855

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4722 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=ghi,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, 7563051);