#7048103

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1209 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,e^2=f,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, 6955299);