#6966428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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)(3,7)(4,6)(5,8)(17,18,23,25,21,27,22,28)(19,29,36,40,31,24,32,38)(20,33,37,42,34,26,35,41)(30,39,46,47,44,43,45,48);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,21,31)(18,24,27,29)(20,30,34,44)(22,32,23,36)(25,38,28,40)(26,39,33,43)(35,45,37,46)(41,47,42,48);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(18,25)(20,34)(22,23)(24,38)(26,42)(27,28)(29,40)(30,44)(32,36)(33,41)(39,48)(43,47);(2,6)(5,7)(10,13)(14,16)(17,20,21,34)(18,26,27,33)(19,30,31,44)(22,35,23,37)(24,39,29,43)(25,41,28,42)(32,45,36,46)(38,47,40,48);(9,11,12,15)(10,14,13,16)(18,27)(19,31)(20,34)(25,28)(32,36)(35,37)(39,43)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873624);