#379132

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

922 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)(9,10,15,20,16,21,14,31)(11,17,13,19,24,33,23,38)(12,25,29,37,30,36,28,18)(22,35,27,40,39,34,26,32);(1,3,5,4)(2,6,8,7)(9,11,15,13,16,24,14,23)(10,17,20,19,21,33,31,38)(12,26,28,39,30,27,29,22)(18,34,36,40,37,35,25,32);(1,3,5,4)(2,7,8,6)(9,12,14,28,16,30,15,29)(10,18,20,25,21,37,31,36)(11,22,23,26,24,39,13,27)(17,32,19,35,33,40,38,34);(1,4,5,3)(2,7,8,6)(9,13,15,24,16,23,14,11)(10,19,20,33,21,38,31,17)(12,27,29,39,30,26,28,22)(18,35,25,40,37,34,36,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286328);