#6545000

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×#3284	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2035 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)(13,14,19,32,18,23,17,22)(15,20,25,34,27,36,26,35)(16,28,29,21,31,37,30,38)(24,39,41,33,42,43,40,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,20,23,36)(16,24,31,42)(17,26,19,25)(21,33,38,44)(22,35,32,34)(28,39,37,43)(29,41,30,40);(6,9)(10,12)(13,16)(14,21)(15,24)(17,29)(18,31)(19,30)(20,33)(22,37)(23,38)(25,40)(26,41)(27,42)(28,32)(34,39)(35,43)(36,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,32)(15,25,27,26)(16,29,31,30)(20,34,36,35)(21,37,38,28)(24,40,42,41)(33,39,44,43);(7,11)(10,12)(13,17,18,19)(14,22,23,32)(15,26,27,25)(16,29,31,30)(20,35,36,34)(21,37,38,28)(24,41,42,40)(33,43,44,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^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, 6452196);