#6558051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2486 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)(13,14,19,23)(15,24,27,34)(16,21,29,37)(17,22,18,32)(20,30,35,26)(25,39,41,44)(28,36,31,38)(33,40,43,42);(5,6)(7,10)(8,9)(11,12)(13,15,17,30,19,27,18,26)(14,20,32,34,23,35,22,24)(16,25,31,40,29,41,28,42)(21,33,36,44,37,43,38,39);(1,2,3,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,28)(19,29)(20,33)(22,38)(23,37)(24,39)(26,42)(27,41)(30,40)(32,36)(34,44)(35,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,26,27,30)(16,28,29,31)(20,34,35,24)(21,36,37,38)(25,40,41,42)(33,39,43,44);(7,11)(10,12)(16,29)(21,37)(25,41)(28,31)(33,43)(36,38)(39,44)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,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, 6465247);