#7574918

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,28)(19,24,32,41)(20,25,21,26)(22,27,37,46)(29,38,34,43)(30,39,47,48)(31,40,36,45)(33,42,35,44);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,34)(22,36)(23,32)(25,38)(26,43)(27,45)(28,41)(30,35)(31,37)(33,47)(39,44)(40,46)(42,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,23)(22,33)(24,38)(26,28)(27,42)(30,31)(32,34)(35,37)(36,47)(39,40)(41,43)(44,46)(45,48);(10,13)(14,16)(17,21,23,20)(18,26,28,25)(19,30,32,47)(22,33,37,35)(24,39,41,48)(27,42,46,44)(29,36,34,31)(38,45,43,40);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(20,21)(25,26)(29,34)(30,47)(33,35)(38,43)(39,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7482114);