#7037490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1237 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6944686);