#496499

order	512
nilpotency class	4
isoclinism	isoclinic class of #481373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403695);