#471704

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

720 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)(9,10,13,11)(12,15,16,14)(17,18,20,24)(19,27,29,22)(21,26,28,25)(23,31,32,30);(1,3,4,7)(2,5,6,8)(10,11)(14,15)(17,19,28,31,20,29,21,30)(18,22,25,23,24,27,26,32);(9,11,13,10)(12,14,16,15)(17,19,20,29)(18,23,24,32)(21,31,28,30)(22,26,27,25);(2,6)(5,8)(9,12,13,16)(10,14,11,15)(18,24)(19,29)(25,26)(30,31)

Polycyclic presentation:

2^9|a^2=e,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, 378900);