#496061

order	512
nilpotency class	4
isoclinism	isoclinic class of #483956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

573 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)(11,21,26,37)(12,27,24,36)(13,29)(14,31)(15,20)(16,25,34,39)(17,28,35,40)(18,30)(19,32)(22,38)(23,33);(2,5)(6,8)(9,11,22,39,14,12,23,40,15,26,13,25,32,24,30,28)(10,16,33,37,19,35,38,27,20,34,18,21,31,17,29,36);(1,3,4,7)(2,6,5,8)(9,12,13,28,14,26,30,39,15,24,22,40,32,11,23,25)(10,17,18,27,19,16,29,21,20,35,33,36,31,34,38,37);(11,24,26,12)(13,22)(16,17,34,35)(18,33)(21,36,37,27)(23,30)(25,28,39,40)(29,38)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=e,d^2=g,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=dh,g^b=gi,d^c=dh,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265093, #72265116, #72265287, …
maximal normal subgroup of	#72265093, #72265116, #72265287, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403257);