#6989506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1181 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,31,33,29,35,30,36)(27,37,43,47,39,32,40,45)(28,34,44,48,41,50,42,49)(38,51,54,56,52,46,53,55);(1,3,7,4)(2,5,8,6)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,32)(28,38)(29,39)(30,40)(31,43)(33,45)(34,46)(35,37)(36,47)(41,52)(42,53)(44,54)(48,55)(49,56)(50,51);(1,4,7,3)(2,5,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(26,33)(28,41)(30,31)(32,45)(34,49)(35,36)(37,47)(38,52)(40,43)(46,56)(48,50)(51,55);(1,5,7,6)(2,3,8,4)(10,16)(14,17)(15,24)(22,23)(25,28)(26,34)(27,38)(29,41)(30,42)(31,44)(32,46)(33,48)(35,50)(36,49)(37,51)(39,52)(40,53)(43,54)(45,55)(47,56);(1,6,7,5)(2,3,8,4)(27,39)(28,41)(32,37)(34,50)(40,43)(42,44)(45,47)(48,49)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=fi,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6896702);