#290940

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198136);