#6644751

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26,31,37)(27,39,43,33)(28,35,29,46)(30,38,32,36)(34,47,52,45)(40,54,41,49)(42,50,44,51)(48,55,53,56);(1,3,4,8)(2,5,6,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,31,43)(26,33,37,39)(28,40,29,41)(30,42,32,44)(34,48,52,53)(35,49,46,54)(36,50,38,51)(45,55,47,56);(3,8)(5,7)(25,28,31,29)(26,34,37,52)(27,40,43,41)(30,45,32,47)(33,48,39,53)(35,38,46,36)(42,55,44,56)(49,51,54,50);(2,6)(5,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,29,31,28)(26,35,37,46)(27,41,43,40)(30,45,32,47)(33,49,39,54)(34,38,52,36)(42,55,44,56)(48,51,53,50);(10,16)(14,17)(15,20)(23,24)(25,30,31,32)(26,36,37,38)(27,42,43,44)(28,45,29,47)(33,50,39,51)(34,35,52,46)(40,55,41,56)(48,49,53,54)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6551947);