#186844

order	512
nilpotency class	3
isoclinism	isoclinic class of #186842
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62833	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94040);