#6597180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2157 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

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, 6504376);