#401704

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 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)(5,6,11,14,10,15,9,25)(7,16,19,29,20,28,18,12)(8,21,26,30,23,34,24,13)(17,27,33,31,22,35,32,36);(5,7,9,18,10,20,11,19)(6,12,14,16,15,29,25,28)(8,22,24,33,23,17,26,32)(13,27,21,31,30,35,34,36);(5,8,9,24,10,23,11,26)(6,13,14,21,15,30,25,34)(7,17,18,32,20,22,19,33)(12,27,16,31,29,35,28,36);(2,4)(8,23)(13,30)(17,22)(21,34)(24,26)(27,35)(31,36)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308900);