#222249

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×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

875 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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, 129445);