#7498606

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

1297 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=i,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=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, 7405802);