#365298

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 272494);