#375766

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1691 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)(9,10,16,21)(11,22,27,35)(12,28,13,32)(14,20,15,19)(17,26,36,25)(18,30,37,31)(23,33,38,39)(24,34,29,40);(9,11,16,27)(10,17,21,36)(12,29,13,24)(14,26,15,25)(18,34,37,40)(19,35,20,22)(23,31,38,30)(28,39,32,33);(2,4)(9,12,16,13)(10,18,21,37)(11,23,27,38)(14,30,15,31)(17,33,36,39)(19,32,20,28)(22,34,35,40)(24,25,29,26);(5,7,8,6)(9,13,16,12)(10,18,21,37)(11,24,27,29)(14,30,15,31)(17,34,36,40)(19,28,20,32)(22,33,35,39)(23,25,38,26)

Polycyclic presentation:

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

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, 282962);