#496729

order	512
nilpotency class	4
isoclinism	isoclinic class of #483439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240836, #72240837, #72240839, …
maximal normal subgroup of	#72240836, #72240837, #72240839, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403925);