#377905

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285101);