#362298

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
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×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

816 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)(3,7)(4,6)(5,8)(9,10,14,18,15,20,13,19)(11,16,23,31,24,33,22,32)(12,25,28,35,29,36,27,17)(21,34,26,39,38,40,37,30);(1,3,5,4)(2,6,8,7)(9,11,13,22,15,24,14,23)(10,16,19,32,20,33,18,31)(12,26,28,38,29,37,27,21)(17,34,25,39,35,40,36,30);(1,3,5,4)(2,7,8,6)(9,12,13,27,15,29,14,28)(10,17,18,25,20,35,19,36)(11,21,22,37,24,38,23,26)(16,30,31,34,33,39,32,40);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

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

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, 269494);