#6483352

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	#30251	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2073 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)(7,12)(8,10)(9,11)(13,14)(15,23)(16,20)(17,30)(18,21)(19,26)(22,31)(24,37)(25,38)(27,42)(28,36)(29,44)(32,40)(33,41)(34,43)(35,39);(7,8)(10,12)(13,15,17,26,18,25,31,41)(14,19,22,23,21,33,30,38)(16,27,29,43,28,24,39,40)(20,34,35,42,36,32,44,37);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16,18,28)(14,20,21,36)(15,24,25,27)(17,29,31,39)(19,32,33,34)(22,35,30,44)(23,37,38,42)(26,40,41,43);(5,8,9,7)(6,10,11,12)(15,25)(19,33)(23,38)(24,27)(26,41)(32,34)(37,42)(40,43);(6,11)(10,12)(14,21)(16,28)(19,33)(22,30)(23,38)(24,27)(29,39)(40,43)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,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, 6390548);