#354875

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 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)(7,13,21,31)(8,24,20,29)(9,15)(10,18)(11,16)(12,17)(14,25,33,26)(19,34)(22,30,23,32)(27,36)(28,35);(5,7,12,23,10,21,11,22)(6,13,17,32,18,31,16,30)(8,19,26,9,20,28,25,27)(14,15,29,35,33,36,24,34);(2,4)(5,8,10,20)(6,14)(7,19,21,28)(9,22,27,23)(11,25,12,26)(13,15)(16,24)(17,29)(18,33)(30,34)(31,36)(32,35);(5,9,12,28,10,27,11,19)(6,15,17,35,18,36,16,34)(7,20,23,25,21,8,22,26)(13,29,32,33,31,24,30,14)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262071);