#464631

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×#3061	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)(13,14,18,23,20,24,19,35)(15,21,27,38,29,39,28,42)(16,30,32,40,17,33,34,22)(25,37,31,43,41,44,26,36);(6,9)(10,12)(13,15,18,27,20,29,19,28)(14,21,23,38,24,39,35,42)(16,31,32,41,17,26,34,25)(22,37,30,43,40,44,33,36);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,20,17)(14,22,24,40)(15,25,29,41)(18,34,19,32)(21,36,39,43)(23,33,35,30)(26,28,31,27)(37,38,44,42);(7,11)(10,12)(13,17,20,16)(14,22,24,40)(15,26,29,31)(18,34,19,32)(21,37,39,44)(23,30,35,33)(25,28,41,27)(36,38,43,42)

Polycyclic presentation:

2^9|a^2=f,b^2=g,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, 371827);