#496500

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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	#72310113, #72310123, #72310125, …
maximal normal subgroup of	#72310113, #72310123, #72310125, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403696);