#6445597

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2114 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)(7,11)(8,9)(10,12)(13,14)(15,20)(16,29)(17,22)(18,23)(19,24)(21,26)(25,40)(27,36)(28,37)(30,44)(31,43)(32,39)(33,38)(34,41)(35,42);(2,4)(6,9)(10,11)(13,15)(14,20)(16,30)(17,32)(18,27)(19,35)(21,38)(22,39)(23,36)(24,42)(25,31)(26,33)(28,34)(29,44)(37,41)(40,43);(5,7,8,12)(6,10,9,11)(13,16,18,26)(14,21,23,29)(15,25,27,17)(19,32,34,31)(20,22,36,40)(24,43,41,39)(28,33,35,30)(37,44,42,38);(13,17,18,25)(14,22,23,40)(15,26,27,16)(19,30,34,33)(20,21,36,29)(24,44,41,38)(28,31,35,32)(37,43,42,39);(15,27)(16,26)(20,36)(21,29)(28,35)(31,32)(37,42)(39,43)

Polycyclic presentation:

2^9|a^2=fgi,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352793);