#7222943

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1225 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7130139);