#7039872

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1210 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)(25,26,30,35)(27,37,39,32)(28,41,29,34)(31,46,45,36)(33,44,50,42)(38,51,43,48)(40,49,55,53)(47,54,52,56);(1,2,5,6)(3,8,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,38)(29,43)(30,39)(31,40)(33,47)(34,51)(35,37)(36,49)(41,48)(42,54)(44,56)(45,55)(46,53)(50,52);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,29)(26,33,35,50)(27,38,39,43)(31,44,45,42)(32,47,37,52)(34,46,41,36)(40,56,55,54)(48,49,51,53);(1,3,5,4)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,29,30,28)(26,34,35,41)(27,38,39,43)(31,44,45,42)(32,48,37,51)(33,46,50,36)(40,54,55,56)(47,49,52,53);(1,4,5,3)(2,7,6,8)(10,16)(11,18)(15,24)(17,19)(28,29)(33,50)(34,41)(38,43)(42,44)(47,52)(48,51)(54,56)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=f,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=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6947068);