#186984

order	512
nilpotency class	3
isoclinism	isoclinic class of #186976
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	#62831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2055 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,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

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239886
maximal normal subgroup of	#72239886
maximal quotient of	#72239898

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94180);