#7646291

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

4748 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,23,25)(19,26,21,24)(20,22,30,32)(27,29,28,31);(9,11,12,16)(10,13,14,15)(17,19,23,21)(18,24,25,26)(20,28,30,27)(22,29,32,31);(1,3,5,4)(2,6,7,8)(17,20)(18,22)(19,27)(21,28)(23,30)(24,31)(25,32)(26,29);(2,7)(6,8)(17,20)(18,22)(19,28)(21,27)(23,30)(24,29)(25,32)(26,31);(1,2)(3,6)(4,8)(5,7)(17,21,23,19)(18,24,25,26)(20,28,30,27)(22,31,32,29);(1,4,5,3)(2,8,7,6)(17,22)(18,20)(19,29)(21,31)(23,32)(24,28)(25,30)(26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7553487);