#421051

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1862 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,e^d=ei,g^d=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, 328247);