#492956

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1240 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)(6,12,14,21)(13,16,28,32)(15,26,30,35)(29,33,31,36);(2,4)(5,6,9,15,10,12,24,26,11,14,25,30,8,21,22,35)(7,16,19,33,20,28,34,31,17,32,23,36,18,13,27,29);(5,7)(6,13,14,28)(8,18)(9,23)(10,20)(11,17)(12,16,21,32)(15,31,30,29)(19,25)(22,34)(24,27)(26,36,35,33);(7,17)(13,28)(16,32)(18,20)(19,23)(27,34)(29,31)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^b=ch,e^b=ei,f^b=fi,d^c=di,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, 400152);