#233919

order	512
nilpotency class	3
isoclinism	isoclinic class of #233907
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	#60676	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^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=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,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, 141115);