#6993671

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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

1312 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,32,37)(27,33,40,51)(28,41,29,35)(30,36,31,46)(34,44,53,43)(38,48,42,56)(39,49,45,50)(47,54,52,55);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27)(26,33)(28,42)(29,38)(30,45)(31,39)(32,40)(34,52)(35,48)(36,50)(37,51)(41,56)(43,54)(44,55)(46,49)(47,53);(3,8)(5,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28)(26,34)(27,38)(29,32)(30,43)(31,44)(33,47)(35,36)(37,53)(39,54)(40,42)(41,46)(45,55)(48,49)(50,56)(51,52);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,29,32,28)(26,35,37,41)(27,38,40,42)(30,43,31,44)(33,48,51,56)(34,46,53,36)(39,55,45,54)(47,50,52,49);(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(25,30,32,31)(26,36,37,46)(27,39,40,45)(28,43,29,44)(33,49,51,50)(34,35,53,41)(38,54,42,55)(47,48,52,56)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fhi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6900867);