#233854

order	512
nilpotency class	3
isoclinism	isoclinic class of #233836
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	C₂×C₄²	C₂⁴
prime radical	#60965	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1201 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:

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

Polycyclic presentation:

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

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, 141050);