#7551018

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#24778	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2071 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458214);