#374262

order	512
nilpotency class	3
isoclinism	isoclinic class of #196332
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 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)(4,5)(6,8)(9,10,14,18,13,17,26,31)(11,19,22,36,21,15,28,30)(12,16,25,33,24,32,27,34)(20,29,37,39,23,35,38,40);(2,5)(6,7)(9,11)(10,15)(12,23)(13,21)(14,22)(16,29)(17,19)(18,30)(20,24)(25,38)(26,28)(27,37)(31,36)(32,35)(33,39)(34,40);(9,12,14,27,13,24,26,25)(10,16,31,33,17,32,18,34)(11,20,28,37,21,23,22,38)(15,29,30,40,19,35,36,39);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,32)(18,31)(29,35)(30,36)(33,34)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318757, #72318775, #72318981, …
maximal normal subgroup of	#72318757, #72318775, #72318981, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281458);