#8105225

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

5022 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)(3,7)(4,6)(5,8)(17,18)(19,27)(20,25)(21,26)(22,24)(23,29)(28,32)(30,31);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,27)(25,32)(26,31);(2,6)(3,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,31)(30,32);(2,6)(3,8)(10,13)(14,16)(18,24)(23,27)(25,26)(31,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,31,27,32);(17,21,22,20)(18,26,24,25)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

2^9|e^2=gi,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	#72304029
maximal normal subgroup of	#72304029
maximal quotient of	#72304041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012421);