#7044518

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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)(27,32)(28,38)(29,34)(30,45)(31,35)(33,40)(36,49)(37,46)(39,55)(41,48)(42,56)(43,51)(44,52)(47,54)(50,53);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,30)(26,32,35,45)(28,36,41,40)(29,37,44,43)(33,38,49,48)(34,46,52,51)(39,53,42,54)(47,55,50,56);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,40,31,41,27,36)(26,33,32,38,35,49,45,48)(29,42,43,53,44,39,37,54)(34,50,46,56,52,47,51,55);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,29,31,44)(26,34,35,52)(27,37,30,43)(28,39,41,42)(32,46,45,51)(33,47,49,50)(36,53,40,54)(38,55,48,56);(10,16)(11,18)(15,24)(17,19)(25,27,31,30)(26,32,35,45)(28,40,41,36)(29,43,44,37)(33,48,49,38)(34,51,52,46)(39,53,42,54)(47,55,50,56)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fg,d^2=fi,e^2=i,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,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6951714);