#7105554

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(30,44)(31,35)(33,36)(37,47)(39,55)(40,46)(41,56)(42,51)(43,52)(45,49)(48,53)(50,54);(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)(26,32,35,44)(28,36,40,45)(29,37,43,42)(33,46,49,38)(34,47,52,51)(39,53,41,54)(48,56,50,55);(1,2,5,4)(3,6,8,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,31,40)(26,33,35,49)(27,36,30,45)(29,41,43,39)(32,46,44,38)(34,50,52,48)(37,54,42,53)(47,55,51,56);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,29,31,43)(26,34,35,52)(27,37,30,42)(28,39,40,41)(32,47,44,51)(33,48,49,50)(36,53,45,54)(38,55,46,56);(10,16)(11,18)(15,24)(17,19)(25,27,31,30)(26,32,35,44)(28,36,40,45)(29,42,43,37)(33,46,49,38)(34,51,52,47)(39,54,41,53)(48,55,50,56)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fhi,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,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, 7012750);