#6993740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
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

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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fhi,d^2=fh,e^2=fh,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,g^c=gi

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, 6900936);