#268042

order	512
nilpotency class	3
isoclinism	isoclinic class of #267947
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61581	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

972 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)(3,9,5,7)(4,12)(6,10)(8,14)(11,16,13,15)(17,18,21,26,23,28,22,27)(19,24,30,39,32,41,31,40)(20,33,37,25,36,44,35,43)(29,45,34,38,47,42,46,48);(1,3,6,5)(2,7,10,9)(4,11,14,13)(8,15,12,16)(17,19,23,32)(18,24,28,41)(20,34,36,46)(21,30,22,31)(25,42,43,45)(26,39,27,40)(29,37,47,35)(33,38,44,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20)(18,25)(19,29)(21,35)(22,37)(23,36)(24,38)(26,33)(27,44)(28,43)(30,46)(31,34)(32,47)(39,45)(40,42)(41,48);(1,5,6,3)(2,9,10,7)(4,13,14,11)(8,16,12,15)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,35,36,37)(24,39,41,40)(25,33,43,44)(29,46,47,34)(38,45,48,42)

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175238);