#186990

order	512
nilpotency class	3
isoclinism	isoclinic class of #186985
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2040 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239886
maximal normal subgroup of	#72239886
maximal quotient of	#72239899

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94186);