#8092471

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5138 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,31)(28,32);(3,7)(5,8)(9,11,14,10)(12,15,13,16)(17,19,22,29)(18,23,25,26)(20,27,21,28)(24,31,30,32);(1,3)(2,5)(4,7)(6,8)(9,12,14,13)(10,15,11,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32);(3,7)(5,8)(9,13,14,12)(10,15,11,16)(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,32,26,31);(2,6)(5,8);(17,21,22,20)(18,24,25,30)(19,28,29,27)(23,31,26,32)

Polycyclic presentation:

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

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, 7999667);