#8092475

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

5002 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,25)(21,26)(22,24)(23,28)(29,32)(30,31);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,19,22,28)(18,23,24,27)(20,29,21,30)(25,31,26,32);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(18,24)(23,27)(25,26)(31,32);(2,7)(6,8)(11,15)(13,16)(18,24)(19,28)(25,26)(29,30);(1,4,5,3)(2,8,7,6)(10,14)(13,16)(17,20,22,21)(18,25,24,26)(19,29,28,30)(23,31,27,32);(17,21,22,20)(18,26,24,25)(19,30,28,29)(23,32,27,31)

Polycyclic presentation:

2^9|b^2=g,e^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, 7999671);