#7198233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=fgh,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105429);