#461756

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1007 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)(5,6,10,9)(7,12,8,11)(13,14,17,19)(15,23,25,40)(16,27,29,39)(18,33,32,22)(20,36,34,26)(21,30,35,31)(24,37,28,38)(41,42,44,43);(1,2,4,3)(5,7,10,8)(6,11,9,12)(14,19)(20,34)(21,35)(22,33)(23,40)(27,39)(37,38)(41,44);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,15,18,26,17,25,32,36)(14,20,22,23,19,34,33,40)(16,28,30,43,29,24,31,42)(21,38,39,44,35,37,27,41);(1,3,4,2)(5,9,10,6)(7,12,8,11)(13,16,17,29)(14,21,19,35)(15,24,25,28)(18,30,32,31)(20,37,34,38)(22,39,33,27)(23,41,40,44)(26,42,36,43)

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=efi,d^2=e,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, 368952);