#7485229

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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1266 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,33,29,35,30,36)(27,37,44,46,38,32,40,48)(28,34,45,49,42,52,43,51)(39,50,54,55,41,47,53,56);(1,2,4,5)(3,7,8,6)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,29,38)(26,32,35,37)(28,41,42,39)(30,40,31,44)(33,48,36,46)(34,50,52,47)(43,54,45,53)(49,55,51,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(27,38)(28,42)(30,31)(32,46)(34,51)(35,36)(37,48)(47,56)(49,52)(50,55)(53,54);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,42)(26,34,35,52)(27,39,38,41)(30,43,31,45)(32,47,37,50)(33,49,36,51)(40,53,44,54)(46,55,48,56);(1,3,4,8)(2,6,5,7)(10,18)(15,19)(16,17)(23,24)(27,38)(28,42)(32,37)(34,52)(40,44)(43,45)(46,48)(49,51)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,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, 7392425);