#6459680

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3096 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)(7,9)(8,10)(11,12)(13,14,19,23)(15,24,25,40)(16,21,31,39)(17,22,18,33)(20,27,34,28)(26,41,29,44)(30,37,32,38)(35,43,36,42);(5,7)(6,9)(8,11)(10,12)(13,15,18,28,19,25,17,27)(14,20,22,24,23,34,33,40)(16,29,30,43,31,26,32,42)(21,36,38,44,39,35,37,41);(1,2,3,4)(6,10)(7,11)(15,25)(20,34)(24,40)(26,29)(27,28)(35,36)(41,44)(42,43);(7,11)(9,12)(13,16)(14,21)(15,26)(17,32)(18,30)(19,31)(20,35)(22,38)(23,39)(24,41)(25,29)(27,43)(28,42)(33,37)(34,36)(40,44);(13,17,19,18)(14,22,23,33)(15,27,25,28)(16,30,31,32)(20,24,34,40)(21,37,39,38)(26,42,29,43)(35,44,36,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366876);