#6513245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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×#3562	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3078 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)(13,14,17,33,19,23,18,22)(15,24,27,20,26,40,28,36)(16,29,34,21,31,39,32,38)(25,35,43,41,30,37,42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,30)(17,28)(18,27)(19,26)(21,37)(22,40)(23,36)(24,33)(25,31)(29,44)(32,42)(34,43)(35,38)(39,41);(6,9)(7,11)(13,16)(14,21)(15,25)(17,32)(18,34)(19,31)(20,35)(22,39)(23,38)(24,41)(26,30)(27,42)(28,43)(29,33)(36,37)(40,44);(5,7,8,11)(6,10,9,12)(15,26)(16,31)(20,36)(21,38)(24,40)(27,28)(29,39)(32,34);(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,27,26,28)(16,32,31,34)(20,24,36,40)(21,39,38,29)(25,42,30,43)(35,41,37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420441);