#8458107

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

4973 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)(3,4,8,7)(10,13)(11,12)(18,20)(19,25)(24,28)(27,29);(2,6)(4,7)(9,10,14,13)(11,16,12,15)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32);(1,3)(2,7)(4,6)(5,8)(9,11,14,12)(10,15,13,16);(1,4,5,7)(2,3,6,8)(9,12,14,11)(10,16,13,15)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31);(9,13,14,10)(11,16,12,15)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31);(1,3)(2,4)(5,8)(6,7)(17,21,22,30)(18,24,20,28)(19,27,25,29)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8365303);