#235206

order	512
nilpotency class	3
isoclinism	isoclinic class of #235187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1235 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142402);