#186166

order	512
nilpotency class	3
isoclinism	isoclinic class of #186162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62838	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2047 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239892
maximal normal subgroup of	#72239892
maximal quotient of	#72239900

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93362);