#220219

order	512
nilpotency class	3
isoclinism	isoclinic class of #220210
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

816 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ci,d^b=de,e^b=eh,d^c=di,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, 127415);