#235202

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

1172 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)(3,12)(4,14)(5,10)(6,16)(7,11)(8,13)(9,15)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,32)(25,36)(30,41)(31,40)(33,39)(35,43)(37,42)(38,46)(44,45)(47,48);(1,3,7,9)(2,8,11,4)(5,13,6,14)(10,15,16,12)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,41)(26,40)(27,29)(28,39)(34,44)(35,46)(36,47)(37,45)(38,42)(43,48);(1,4,7,8)(2,9,11,3)(5,15,6,12)(10,13,16,14)(17,20,23,35)(18,25,28,42)(19,30,31,46)(21,36,22,37)(24,38,39,41)(26,34,27,43)(29,44,40,48)(32,45,33,47);(1,5,7,6)(2,10,11,16)(3,13,9,14)(4,12,8,15)(19,31)(20,35)(24,39)(25,42)(29,40)(32,33)(34,43)(36,37)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,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, 142398);