#7492795

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,31,33,29,35,30,36)(27,37,45,49,39,32,40,47)(28,41,46,52,43,34,44,50)(38,48,54,55,42,51,53,56);(2,6)(3,4)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,29,39)(26,32,35,37)(28,42,43,38)(30,40,31,45)(33,47,36,49)(34,51,41,48)(44,54,46,53)(50,56,52,55);(1,2)(3,8)(4,7)(5,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(30,31)(32,47)(34,50)(35,36)(37,49)(40,45)(41,52)(44,46)(48,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,43)(26,34,35,41)(27,38,39,42)(30,44,31,46)(32,48,37,51)(33,50,36,52)(40,53,45,54)(47,55,49,56);(1,4,5,3)(2,7,6,8)(10,18)(15,19)(16,17)(23,24)(27,39)(28,43)(32,37)(34,41)(40,45)(44,46)(47,49)(50,52)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7399991);