#6487715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#32913	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 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)(7,10)(8,11)(9,12)(13,14)(15,20)(16,27)(17,22)(18,31)(19,23)(21,28)(24,39)(25,34)(26,35)(29,37)(30,36)(32,38)(33,42)(40,44)(41,43);(1,2,3,4)(7,8)(10,11)(13,15)(14,20)(16,24)(17,30)(18,26)(19,25)(21,33)(22,36)(23,34)(27,39)(28,42)(29,40)(31,35)(32,41)(37,44)(38,43);(5,7,9,8)(6,10,12,11)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,37)(23,38)(25,40)(26,41)(27,31)(30,42)(34,43)(35,39)(36,44);(5,8,9,7)(6,11,12,10)(15,25)(20,34)(24,40)(26,30)(33,43)(35,36)(39,44)(41,42);(6,12)(10,11)(13,17,19,18)(14,22,23,31)(15,26,25,30)(16,28,29,32)(20,35,34,36)(21,37,38,27)(24,41,40,42)(33,39,43,44)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi

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, 6394911);