#460116

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1193 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 367312);