#6425688

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#52096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=dfi,e^c=ei

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, 6332884);