#464635

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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	C₂×#3060	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1518 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,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^c=de,f^c=fi,g^c=gi,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, 371831);