#6792983

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1343 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)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,29,34)(27,36,38,31)(28,40,41,33)(30,32,43,35)(37,45,50,49)(39,46,52,44)(42,48,53,47)(51,54,56,55);(1,2)(3,7)(4,5)(6,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27)(26,31)(28,37)(29,38)(30,39)(32,44)(33,45)(34,36)(35,46)(40,49)(41,50)(42,51)(43,52)(47,54)(48,55)(53,56);(2,5)(6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(26,32)(30,43)(31,44)(33,47)(34,35)(36,46)(39,52)(40,48)(42,53)(45,54)(49,55)(51,56);(10,17)(15,19)(16,18)(23,24)(25,28,29,41)(26,33,34,40)(27,37,38,50)(30,42,43,53)(31,45,36,49)(32,47,35,48)(39,51,52,56)(44,54,46,55);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,38)(28,41)(31,36)(33,40)(39,52)(42,53)(44,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6700179);