#6963687

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^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, 6870883);