#6963552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 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)(9,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,29,32,24)(20,33,35,40)(21,28,23,26)(25,34,41,36)(30,44,46,48)(31,39,37,42)(38,45,43,47);(2,6)(4,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,30)(21,37)(22,32)(23,31)(25,38)(26,42)(27,29)(28,39)(33,48)(34,47)(35,46)(36,45)(40,44)(41,43);(1,3)(2,7)(4,6)(5,8)(11,16)(13,15)(17,20,22,35)(18,25,27,41)(19,30,32,46)(21,36,23,34)(24,38,29,43)(26,33,28,40)(31,47,37,45)(39,44,42,48);(2,6)(3,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,34,35,36)(24,39,29,42)(25,40,41,33)(30,45,46,47)(38,48,43,44);(1,4,5,7)(2,3,6,8)(18,27)(19,32)(25,41)(26,28)(30,46)(31,37)(33,40)(45,47)

Polycyclic presentation:

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

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, 6870748);