#421972

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1052 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,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, 329168);