#7489590

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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

1328 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)(27,36)(28,33)(29,34)(30,32)(31,37)(35,42)(38,49)(39,43)(40,48)(41,46)(44,50)(45,51)(47,53)(52,54)(55,56);(2,6)(3,4)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,38)(29,37)(30,39)(32,45)(33,44)(34,36)(35,43)(40,50)(41,52)(42,51)(46,55)(47,54)(48,49)(53,56);(1,2)(3,8)(4,7)(5,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,37)(28,40)(30,42)(31,43)(33,47)(34,35)(36,45)(44,55)(46,48)(49,54)(52,56);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,40)(26,33,34,48)(27,38,37,50)(30,41,42,53)(31,44,36,49)(32,46,35,47)(39,52,51,56)(43,54,45,55);(1,4,5,3)(2,7,6,8)(10,18)(15,19)(16,17)(23,24)(26,34)(27,37)(28,40)(32,35)(39,51)(41,53)(44,49)(54,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fhi,e^2=hi,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=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7396786);