#6502892

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	#37445	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2077 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)(20,28)(25,38)(26,39)(27,34)(29,35)(30,37)(31,36)(33,41)(40,44)(42,43);(5,6)(7,11)(8,10)(9,12)(13,15,18,28,19,26,17,27)(14,20,22,24,23,34,32,39)(16,25,29,41,30,40,31,42)(21,33,36,38,37,43,35,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,30)(14,21,23,37)(15,25,26,40)(17,31,18,29)(20,33,34,43)(22,36,32,35)(24,38,39,44)(27,42,28,41);(7,12)(9,11)(15,26)(20,34)(24,39)(25,40)(27,28)(33,43)(38,44)(41,42);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,27,26,28)(16,29,30,31)(20,24,34,39)(21,35,37,36)(25,41,40,42)(33,44,43,38)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,e^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, 6410088);