#7306850

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

1301 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)(25,26,31,37)(27,39,43,33)(28,44,29,35)(30,38,32,36)(34,46,51,45)(40,49,41,54)(42,50,47,52)(48,56,53,55);(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)(26,33)(28,40)(29,41)(30,47)(31,43)(32,42)(34,48)(35,49)(36,52)(37,39)(38,50)(44,54)(45,56)(46,55)(51,53);(1,3)(2,5)(4,8)(6,7)(25,28,31,29)(26,34,37,51)(27,40,43,41)(30,45,32,46)(33,48,39,53)(35,36,44,38)(42,55,47,56)(49,52,54,50);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,29,31,28)(26,35,37,44)(27,41,43,40)(30,45,32,46)(33,49,39,54)(34,36,51,38)(42,55,47,56)(48,52,53,50);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,30,31,32)(26,36,37,38)(27,42,43,47)(28,45,29,46)(33,50,39,52)(34,44,51,35)(40,55,41,56)(48,49,53,54)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,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, 7214046);