#6782801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1194 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,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, 6689997);