#6993871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1344 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fhi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,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, 6901067);