#7604320

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=h,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, 7511516);