#6954404

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

1306 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)(25,26,31,35)(27,32,37,47)(28,39,41,52)(29,34,30,45)(33,42,51,43)(36,53,40,56)(38,48,44,49)(46,55,50,54);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,31,37)(26,32,35,47)(28,40,41,36)(29,44,30,38)(33,50,51,46)(34,49,45,48)(39,56,52,53)(42,54,43,55);(3,8)(5,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,28,31,41)(26,33,35,51)(27,36,37,40)(29,43,30,42)(32,46,47,50)(34,52,45,39)(38,55,44,54)(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=fi,b^2=fi,c^2=fh,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, 6861600);