#6491753

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2047 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)(17,22)(18,33)(19,23)(20,26)(25,40)(27,35)(28,42)(29,38)(30,39)(31,44)(32,36)(34,41)(37,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,27,17,32)(14,20,22,24,23,36,33,35)(16,28,31,43,30,25,29,41)(21,37,38,42,39,34,44,40);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,31)(19,30)(20,34)(22,38)(23,39)(24,40)(26,41)(27,28)(32,43)(33,44)(35,42)(36,37);(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,26,27,32)(16,29,30,31)(20,35,36,24)(21,38,39,44)(25,41,28,43)(34,42,37,40);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(16,30)(22,33)(26,32)(29,31)(34,37)(40,42)

Polycyclic presentation:

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

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, 6398949);