#7500611

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=g,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7407807);