#267998

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	C₂×#553	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

832 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=i,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, 175194);