#6343594

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343584
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,f^2=i,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250790);