#6500428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2746 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei

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, 6407624);