#377005

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 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)(3,8)(4,6)(5,7)(9,10,16,21,14,20,15,35)(11,22,26,37,25,17,36,38)(12,24,30,39,28,40,29,18)(13,31,34,23,32,19,33,27);(1,3,5,4)(2,6,7,8)(9,11,14,25)(10,17,20,22)(12,27,28,23)(13,24,32,40)(15,36,16,26)(18,34,39,33)(19,30,31,29)(21,38,35,37);(1,4,5,3)(2,6,7,8)(9,12,16,30,14,28,15,29)(10,18,35,40,20,39,21,24)(11,23,36,31,25,27,26,19)(13,17,34,38,32,22,33,37);(2,7)(6,8)(9,13,16,34,14,32,15,33)(10,19,21,27,20,31,35,23)(11,24,26,39,25,40,36,18)(12,17,30,38,28,22,29,37)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284201);