#222251

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

894 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)(3,7,13,16,11,14,12,15)(17,18,22,27,23,28,21,37)(19,29,32,24,33,43,38,42)(20,26,35,45,36,25,39,46)(30,41,48,40,34,47,31,44);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,19,23,33)(18,24,28,42)(20,34,36,30)(21,38,22,32)(25,44,26,40)(27,43,37,29)(31,39,48,35)(41,45,47,46);(3,11)(7,14)(12,13)(15,16)(17,20)(18,25)(19,30)(21,35)(22,39)(23,36)(24,40)(26,28)(27,45)(29,47)(31,32)(33,34)(37,46)(38,48)(41,43)(42,44);(1,4,6,5)(2,8,10,9)(3,12,11,13)(7,15,14,16)(17,20)(18,26)(19,31)(21,39)(22,35)(23,36)(24,41)(25,28)(27,45)(29,44)(30,32)(33,48)(34,38)(37,46)(40,43)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129447);