#6812364

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1229 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,38)(28,33)(29,37)(30,35)(31,36)(32,41)(34,44)(39,47)(40,53)(42,49)(43,52)(45,56)(46,51)(48,54)(50,55);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,41)(26,32,36,38)(28,29,39,44)(30,45,46,42)(33,34,47,37)(35,49,51,56)(40,43,54,55)(48,52,53,50);(2,6)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,29,41,31,39,44,27)(26,33,37,32,36,47,34,38)(30,42,43,54,46,45,55,40)(35,49,52,48,51,56,50,53);(10,16)(14,17)(15,22)(21,23)(25,29,31,44)(26,34,36,37)(27,39,41,28)(30,43,46,55)(32,47,38,33)(35,50,51,52)(40,45,54,42)(48,56,53,49);(11,19)(14,17)(18,24)(21,23)(25,30,44,55,31,46,29,43)(26,35,34,50,36,51,37,52)(27,40,39,45,41,54,28,42)(32,48,33,49,38,53,47,56)

Polycyclic presentation:

2^9|b^2=gi,c^2=fhi,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719560);