#6954017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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)(10,14)(12,19)(15,21)(17,20)(25,26,29,34)(27,36,37,31)(28,40,41,33)(30,32,43,35)(38,45,50,49)(39,44,51,46)(42,48,53,47)(52,55,56,54);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27)(26,31)(28,38)(29,37)(30,39)(32,46)(33,45)(34,36)(35,44)(40,49)(41,50)(42,52)(43,51)(47,55)(48,54)(53,56);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(26,32)(27,37)(30,43)(31,44)(33,47)(34,35)(36,46)(38,50)(40,48)(42,53)(45,54)(49,55);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,28)(26,33)(27,38)(29,41)(30,42)(31,45)(32,47)(34,40)(35,48)(36,49)(37,50)(39,52)(43,53)(44,54)(46,55)(51,56);(10,14)(15,21)(16,22)(23,24)(27,37)(28,41)(31,36)(33,40)(39,51)(42,53)(44,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6861213);