#8101213

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

5013 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304031
maximal normal subgroup of	#72304031
maximal quotient of	#72304057

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8008409);