#7304547

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,36)(27,38,42,32)(28,44,46,34)(29,37,31,35)(33,47,52,45)(39,53,40,49)(41,50,43,51)(48,55,54,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,39)(29,41)(30,42)(31,43)(33,48)(34,53)(35,50)(36,38)(37,51)(40,46)(44,49)(45,55)(47,56)(52,54);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,39)(29,45)(30,46)(31,47)(32,48)(34,37)(35,44)(36,52)(38,54)(40,42)(41,55)(43,56)(49,50)(51,53);(2,6)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,40)(29,47)(30,46)(31,45)(32,49)(33,37)(35,52)(36,44)(38,53)(39,42)(41,55)(43,56)(48,50)(51,54);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,35,36,37)(27,41,42,43)(28,45,46,47)(32,50,38,51)(33,44,52,34)(39,55,40,56)(48,49,54,53)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 7211743);