#6835972

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

1339 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,38)(28,42)(29,37)(30,35)(31,36)(32,40)(33,41)(34,43)(39,52)(44,56)(45,51)(46,50)(47,54)(48,55)(49,53);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,29)(30,44)(31,40)(33,37)(34,42)(35,48)(36,38)(39,53)(41,43)(45,54)(46,55)(47,49)(50,56)(51,52);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,43,40,31,41,29,27)(26,33,34,38,36,42,37,32)(30,44,53,54,46,55,45,39)(35,48,49,52,50,56,51,47);(10,16)(14,17)(15,22)(21,23)(25,29,31,43)(26,34,36,37)(27,28,40,41)(30,45,46,53)(32,42,38,33)(35,49,50,51)(39,44,54,55)(47,56,52,48);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,30,29,45,31,46,43,53)(26,35,37,51,36,50,34,49)(27,39,41,55,40,54,28,44)(32,47,42,56,38,52,33,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=fghi,d^2=i,e^2=ghi,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, 6743168);