#362024

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1426 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,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, 269220);