#405068

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

905 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,c^2=fi,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,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, 312264);