#471684

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378880);