#8092922

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

5010 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=gi,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, 8000118);