#7306795

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7213991);