#403224

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

911 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,c^2=fi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,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, 310420);