#382415

order	512
nilpotency class	3
isoclinism	isoclinic class of #184985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3494 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)(9,10,15,20)(11,16,24,33)(12,25,28,36)(13,19,14,18)(17,29,35,27)(21,34,37,40)(22,32,23,31)(26,30,38,39);(2,5)(6,8)(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,34)(18,32)(19,31)(20,33)(21,27)(25,30)(28,38)(29,37)(35,40)(36,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21,24,37)(13,29)(14,27)(15,28)(16,30,33,39)(18,36)(19,25)(20,35)(22,38,23,26)(31,40,32,34);(3,7)(6,8)

Polycyclic presentation:

2^9|a^2=i,c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=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, 289611);