#6435556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^c=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, 6342752);