#439540

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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

1131 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)(5,6,9,11)(7,10,8,12)(13,14)(15,26)(16,32)(17,22)(18,24)(19,25)(20,36)(21,29)(23,35)(27,37)(28,39)(30,42)(31,38)(33,40)(34,44)(41,43);(2,3)(5,7,9,8)(6,10,11,12)(13,15,19,30,18,29,20,31)(14,21,25,38,24,26,36,42)(16,28,33,43,17,34,35,27)(22,39,23,41,32,44,40,37);(1,2,4,3)(5,8,9,7)(6,10,11,12)(13,16,18,17)(14,22,24,32)(15,27,29,43)(19,35,20,33)(21,37,26,41)(23,25,40,36)(28,30,34,31)(38,44,42,39);(1,3,4,2)(13,17,18,16)(14,23,24,40)(15,28,29,34)(19,35,20,33)(21,37,26,41)(22,25,32,36)(27,30,43,31)(38,39,42,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=efi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^c=gi,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, 346736);