#464951

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1165 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,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, 372147);