#8093283

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	#36624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4922 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000479);