#8101210

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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	#72303333
maximal normal subgroup of	#72303333
maximal quotient of	#72303412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8008406);