#359108

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266304);