#6954384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1308 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)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26)(27,32)(28,39)(29,34)(30,45)(31,35)(33,43)(36,53)(37,47)(38,48)(40,56)(41,52)(42,51)(44,49)(46,54)(50,55);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,32)(28,40)(29,44)(30,38)(31,37)(33,50)(34,49)(35,47)(36,41)(39,56)(42,54)(43,55)(45,48)(46,51)(52,53);(3,8)(5,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,28)(26,33)(27,36)(29,43)(30,42)(31,41)(32,46)(34,52)(35,51)(37,40)(38,55)(39,45)(44,54)(47,50)(48,53)(49,56);(2,6)(5,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(27,37)(28,41)(32,47)(33,51)(38,44)(39,52)(42,43)(48,49);(11,19)(15,17)(18,24)(22,23)(25,29,31,30)(26,34,35,45)(27,38,37,44)(28,42,41,43)(32,48,47,49)(33,39,51,52)(36,54,40,55)(46,56,50,53)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fhi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi

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, 6861580);