#238068

order	512
nilpotency class	3
isoclinism	isoclinic class of #238065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

844 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,24)(20,32,34,43)(21,27)(22,37)(23,26)(25,35,42,36)(28,41,44,48)(29,40)(30,39)(31,46)(33,38,45,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,33,34,45)(21,30)(22,31)(23,29)(25,41,42,48)(26,40)(27,39)(28,36,44,35)(32,47,43,38)(37,46);(4,11)(6,12)(7,14)(9,15)(17,20)(18,25)(19,28)(21,36)(22,35)(23,34)(24,38)(26,42)(27,43)(29,44)(30,33)(31,45)(32,37)(39,48)(40,47)(41,46);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(18,26)(19,29)(25,42)(27,37)(28,44)(30,31)(32,43)(33,45)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145264);