#8093072

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

4962 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)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32);(1,4,5,8)(2,6,3,7)(10,14)(11,15)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32);(2,3)(6,7)(11,15)(13,16)(18,25)(20,28)(23,26)(27,29);(18,25)(23,26)(24,31)(30,32);(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)

Polycyclic presentation:

2^9|b^2=g,c^2=g,f^2=ghi,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, 8000268);