#7297369

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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

1282 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,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, 7204565);