#7487539

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1387 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=f,e^2=h,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, 7394735);