#187426

order	512
nilpotency class	3
isoclinism	isoclinic class of #187419
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2021 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238860, C₂×#187426
maximal normal subgroup of	#72238860, C₂×#187426
maximal quotient of	C₂×#187426, #72238884, #72238887

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94622);