#6916284

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=fi,e^2=i,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, 6823480);