#401702

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	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1142 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)(5,6,11,14,10,15,9,24)(7,16,25,28,19,31,18,12)(8,20,26,34,22,13,23,29)(17,30,36,33,21,35,32,27);(5,7,9,18,10,19,11,25)(6,12,14,16,15,28,24,31)(8,21,23,36,22,17,26,32)(13,27,29,30,20,33,34,35);(5,8,9,23,10,22,11,26)(6,13,14,29,15,20,24,34)(7,17,18,32,19,21,25,36)(12,27,16,30,28,33,31,35);(2,4)(8,22)(13,20)(17,21)(23,26)(27,33)(29,34)(30,35)(32,36)

Polycyclic presentation:

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

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, 308898);