#6544789

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×#3282	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2052 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ei,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, 6451985);