#425804

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

1317 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)(7,10)(8,12)(9,11)(13,14,18,24)(15,26,29,37)(16,31,34,21)(17,25,19,23)(20,30,22,28)(27,42,44,38)(32,39,43,36)(33,40,35,41);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,19,30,18,29,17,28)(14,20,25,37,24,22,23,26)(16,32,35,44,34,43,33,27)(21,38,41,39,31,42,40,36);(5,8,9,7)(6,10,11,12)(13,16,18,34)(14,21,24,31)(15,27,29,44)(17,35,19,33)(20,36,22,39)(23,41,25,40)(26,38,37,42)(28,32,30,43);(2,4)(13,15,18,29)(14,22,24,20)(16,27,34,44)(17,30,19,28)(21,39,31,36)(23,26,25,37)(32,35,43,33)(38,40,42,41)

Polycyclic presentation:

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

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, 333000);