#8472950

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

5347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8380146);