#6344016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343312
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251212);