#189560

order	512
nilpotency class	3
isoclinism	isoclinic class of #189531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 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:

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

Polycyclic presentation:

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

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, 96756);