#424314

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1522 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)(5,6,10,15,11,16,23,31)(7,17,20,30,21,34,27,12)(8,22,9,14,24,13,26,32)(18,35,36,29,19,33,25,28);(5,7)(6,12)(8,18)(9,25)(10,27)(11,21)(13,29)(14,33)(15,34)(16,30)(17,31)(19,24)(20,23)(22,28)(26,36)(32,35);(5,8,23,26,11,24,10,9)(6,13,31,14,16,22,15,32)(7,18,20,36,21,19,27,25)(12,28,17,35,30,29,34,33);(2,4)(5,9,10,24,11,26,23,8)(6,14,15,13,16,32,31,22)(7,19,20,25,21,18,27,36)(12,29,17,33,30,28,34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331510);