#189000

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

831 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,e^2=h,f^2=h,g^2=i,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, 96196);