#368655

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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)(5,6,8,7)(9,10)(11,22)(12,17)(13,30)(14,20)(15,21)(16,25)(18,29)(19,23)(24,35)(26,34)(27,37)(28,40)(31,36)(32,38)(33,39);(5,7,8,6)(9,11,23,26,15,27,14,25)(10,16,20,37,21,34,19,22)(12,28,31,39,29,38,13,24)(17,35,30,32,18,33,36,40);(2,4)(9,12,14,13,15,29,23,31)(10,17,20,30,21,18,19,36)(11,24,26,28,27,39,25,38)(16,32,22,35,34,40,37,33);(9,13,14,29,15,31,23,12)(10,18,19,30,21,17,20,36)(11,24,25,38,27,39,26,28)(16,33,22,32,34,35,37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=efhi,c^2=g,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275851);