#356668

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1630 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=de,e^d=ei,h^d=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263864);