#424811

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1279 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:

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

Polycyclic presentation:

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

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, 332007);