#385525

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1074 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)(3,5)(4,7)(6,8)(9,10,15,21,14,19,16,20)(11,22,12,25,24,17,27,34)(13,28,31,36,30,37,32,18)(23,38,26,33,29,40,39,35);(1,3)(2,4)(5,8)(6,7)(9,11)(10,17)(12,15)(13,29)(14,24)(16,27)(18,35)(19,22)(20,25)(21,34)(23,30)(26,32)(28,38)(31,39)(33,36)(37,40);(1,4,6,5)(2,7,8,3)(9,12,16,11,14,27,15,24)(10,17,21,34,19,22,20,25)(13,26,32,23,30,39,31,29)(18,33,28,40,36,35,37,38);(1,5,6,4)(2,7,8,3)(9,13,16,32,14,30,15,31)(10,18,21,28,19,36,20,37)(11,23,27,39,24,29,12,26)(17,33,34,40,22,35,25,38)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292721);