#6993668

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

1310 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)(25,26,30,36)(27,37,41,32)(28,43,45,34)(29,35,31,47)(33,46,52,44)(38,53,39,49)(40,54,42,50)(48,55,51,56);(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)(26,32)(28,39)(29,40)(30,41)(31,42)(33,51)(34,49)(35,50)(36,37)(38,45)(43,53)(44,56)(46,55)(47,54)(48,52);(3,8)(5,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28)(26,33)(27,38)(29,44)(30,45)(31,46)(32,48)(34,35)(36,52)(37,51)(39,41)(40,55)(42,56)(43,47)(49,54)(50,53);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,28)(26,34)(27,39)(29,46)(30,45)(31,44)(32,49)(33,35)(36,43)(37,53)(38,41)(40,55)(42,56)(47,52)(48,54)(50,51);(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(25,29,30,31)(26,35,36,47)(27,40,41,42)(28,44,45,46)(32,50,37,54)(33,34,52,43)(38,55,39,56)(48,53,51,49)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fh,d^2=fh,e^2=fhi,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, 6900864);