#6889606

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,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,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, 6796802);