#425020

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1113 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)(6,9)(10,12)(13,14,17,23,19,24,18,21)(15,25,27,37,29,36,28,20)(16,30,32,22,33,38,34,40)(26,42,31,41,43,35,44,39);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,24,37)(16,31,33,44)(17,28,18,27)(21,25,23,36)(22,39,40,41)(26,34,43,32)(30,42,38,35);(5,7,8,11)(6,10,9,12)(14,21)(17,18)(20,25)(22,38)(23,24)(27,28)(30,40)(31,44)(32,34)(35,41)(36,37)(39,42);(2,4)(6,9)(10,12)(13,16)(14,22)(15,26)(17,34)(18,32)(19,33)(20,35)(21,38)(23,30)(24,40)(25,41)(27,44)(28,31)(29,43)(36,39)(37,42)

Polycyclic presentation:

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

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, 332216);