#6483425

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390621);