#8102659

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

4989 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=g,e^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	#72304029
maximal normal subgroup of	#72304029
maximal quotient of	#72304055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8009855);