#366811

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,27)(11,16,21,32,22,33,28,31)(12,23,25,17,26,35,29,34)(20,36,38,40,24,39,37,30);(2,5)(6,8)(9,11)(10,16)(12,24,26,20)(13,28)(14,21)(15,22)(17,30,34,40)(18,32)(19,33)(23,39,35,36)(25,37,29,38)(27,31);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,29)(14,25)(15,26)(16,30)(18,35)(19,34)(21,38)(22,24)(23,27)(28,37)(31,39)(32,36)(33,40);(2,5)(6,8)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274007);