#441036

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=ei,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348232);