#460901

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

978 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=e,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, 368097);