#7331453

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1321 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fg,d^2=f,e^2=i,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, 7238649);