#238304

order	512
nilpotency class	3
isoclinism	isoclinic class of #238303
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
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

781 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)(4,12)(5,9)(7,11)(8,14)(10,15)(13,16)(17,18)(19,24)(20,32,35,43)(21,26)(22,27)(23,28)(25,34,41,37)(29,44,33,47)(30,39)(31,40)(36,42)(38,46,48,45);(1,3)(2,6)(4,11)(5,10)(7,12)(8,16)(9,15)(13,14)(17,19)(18,24)(20,33,35,29)(21,36)(22,30)(23,31)(25,38,41,48)(26,42)(27,39)(28,40)(32,44,43,47)(34,45,37,46);(1,3)(2,7)(4,13)(5,10)(6,12)(8,15)(9,16)(11,14)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,32)(27,43)(28,41)(30,45)(31,33)(36,46)(39,47)(40,48)(42,44);(3,10)(6,15)(7,16)(11,13)(17,21,23,22)(18,26,28,27)(19,30,31,36)(20,34,35,37)(24,39,40,42)(25,32,41,43)(29,45,33,46)(38,47,48,44)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=h,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,g^b=gh,d^c=di,e^c=eh,f^c=fhi

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, 145500);