#496498

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

878 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)(9,10,14,18,24,32,27,40,12,16,13,25,15,19,29,36)(11,20,22,34,30,39,28,17,21,33,38,37,23,35,26,31);(1,2)(3,6)(4,5)(7,8)(10,16)(11,21)(18,25)(19,32)(22,38)(23,30)(26,28)(36,40);(2,5)(6,8)(9,11)(10,17)(12,21)(13,26)(14,28)(15,30)(16,31)(18,35)(19,37)(20,36)(22,27)(23,24)(25,39)(29,38)(32,34)(33,40);(1,3,4,7)(2,6,5,8)(10,16)(17,31)(18,25)(19,32)(20,33)(34,37)(35,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,d^2=ei,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.

Relationships

maximal subgroup of	#72265287, #72265292, #72265307, …
maximal normal subgroup of	#72265287, #72265292, #72265307, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403694);