#233847

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1128 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:

(1,2,7,10)(3,11,14,15)(4,13,16,12)(5,8,6,9)(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);(1,3,7,14)(2,4,10,16)(5,13,6,12)(8,11,9,15)(17,19,23,32)(18,24,28,40)(20,34,36,47)(21,33,22,31)(25,41,44,45)(26,29,27,39)(30,37,46,35)(38,42,48,43);(1,4)(2,3)(5,15)(6,11)(7,16)(8,12)(9,13)(10,14)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,43)(28,44)(29,41)(31,34)(32,46)(33,47)(39,45)(40,48);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,15,16,11)(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|a^2=i,b^2=h,d^2=h,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, 141043);