#6544978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#3285	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2017 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)(13,14,18,23,19,24,17,22)(15,20,28,36,26,34,27,35)(16,29,30,37,31,38,32,21)(25,39,41,43,40,44,42,33);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,35)(23,36)(24,34)(29,39)(30,41)(31,40)(32,42)(37,43)(38,44);(6,9)(10,12)(13,16,19,31)(14,21,24,37)(15,25,26,40)(17,30,18,32)(20,33,34,43)(22,29,23,38)(27,41,28,42)(35,39,36,44);(5,7)(6,10)(8,11)(9,12)(15,26)(20,34)(25,40)(27,28)(33,43)(35,36)(39,44)(41,42);(7,11)(10,12)(13,17,19,18)(14,22,24,23)(15,27,26,28)(16,30,31,32)(20,35,34,36)(21,29,37,38)(25,41,40,42)(33,39,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452174);