#6452927

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3136 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6360123);