#455973

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1190 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)(14,19)(15,23)(17,30)(20,34)(21,37)(22,31)(25,40)(26,35)(27,43)(28,32)(33,39)(38,41);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,18,22)(15,24,26,42)(16,21,29,39)(17,31,30,19)(20,35,34,23)(25,41,43,38)(27,44,40,36)(28,33,32,37);(2,4)(5,7,8,11)(6,10,9,12)(13,15,17,23,18,26,30,35)(14,20,19,24,22,34,31,42)(16,27,28,43,29,40,32,25)(21,38,37,44,39,41,33,36);(7,11)(10,12)(13,16,18,29)(14,21,22,39)(15,25,26,43)(17,32,30,28)(19,33,31,37)(20,36,34,44)(23,40,35,27)(24,41,42,38)

Polycyclic presentation:

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

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, 363169);