#222846

order	512
nilpotency class	3
isoclinism	isoclinic class of #222720
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fhi,g^c=gh,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130042);