#264657

order	512
nilpotency class	3
isoclinism	isoclinic class of #264619
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61213	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

881 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)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,27)(20,31)(21,35)(22,26)(23,29)(24,33)(25,32)(28,41)(30,39)(34,45)(36,42)(37,44)(38,46)(40,43)(47,48);(2,7)(3,12)(5,6)(8,11)(13,14)(15,16)(17,19,21,29,22,30,32,42)(18,23,25,27,26,36,35,39)(20,28,33,43,34,44,46,48)(24,37,31,40,38,41,45,47);(1,3,4,13)(2,8,7,15)(5,12,6,14)(9,16,10,11)(23,36)(27,39)(37,41)(40,47);(2,7)(8,15)(9,10)(11,16)(17,20,32,46,22,34,21,33)(18,24,35,45,26,38,25,31)(19,28,42,48,30,44,29,43)(23,37,39,47,36,41,27,40)

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171853);