#189032

order	512
nilpotency class	3
isoclinism	isoclinic class of #188956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

829 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)(17,18,22,26,23,28,21,27)(19,29,32,40,33,39,31,24)(20,34,36,25,35,42,37,41)(30,44,46,43,45,38,47,48);(17,19,21,31,23,33,22,32)(18,24,26,29,28,40,27,39)(20,30,36,46,35,45,37,47)(25,38,34,43,41,44,42,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,38)(26,34)(27,42)(28,41)(29,43)(31,46)(32,47)(33,45)(39,48)(40,44);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(20,35)(25,41)(30,45)(34,42)(36,37)(38,44)(43,48)(46,47)

Polycyclic presentation:

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

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, 96228);