#368726

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1740 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=i,f^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, 275922);