#428086

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1183 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=ef,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,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335282);