#496369

order	512
nilpotency class	4
isoclinism	isoclinic class of #481373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1168 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)(3,7)(4,6)(5,8)(9,10,14,19,12,17,26,20)(11,21,24,34,22,15,30,33)(13,27,29,18,25,16,28,36)(23,35,39,37,38,31,40,32);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,22)(13,23)(14,24)(16,35)(17,21)(18,32)(19,33)(20,34)(25,38)(26,30)(27,31)(28,40)(29,39)(36,37);(3,8)(5,7)(10,16)(11,22)(13,28)(14,26)(15,31)(17,27)(18,19)(20,36)(21,35)(23,39)(25,29)(32,34)(33,37)(38,40);(2,6)(5,7)(10,17)(15,21)(16,27)(18,36)(19,20)(31,35)(32,37)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^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,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265283, #72265287, #72265315, …
maximal normal subgroup of	#72265283, #72265287, #72265315, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403565);