#434673

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,20,26)(15,27,32,39)(16,33,36,42)(17,23,28,38)(18,25,19,24)(21,31,40,30)(22,35,43,29)(34,44,37,41);(13,15,18,30,20,32,19,31)(14,21,24,27,26,40,25,39)(16,34,29,17,36,37,35,28)(22,41,33,23,43,44,42,38);(5,7,8,12)(6,9,10,11)(13,16,19,35,20,36,18,29)(14,22,25,42,26,43,24,33)(15,28,30,34,32,17,31,37)(21,38,27,41,40,23,39,44);(2,4)(13,17,20,28)(14,23,26,38)(15,29,32,35)(16,31,36,30)(18,34,19,37)(21,33,40,42)(22,39,43,27)(24,41,25,44)

Polycyclic presentation:

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

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, 341869);