#238427

order	512
nilpotency class	3
isoclinism	isoclinic class of #238422
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

766 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,23)(20,30,32,40)(21,25)(22,26)(24,33,39,34)(27,42,31,46)(28,37)(29,38)(35,41)(36,43,47,45)(44,48);(4,5)(7,15)(8,14)(11,12)(17,19)(18,23)(20,31,32,27)(21,29)(22,28)(24,36,39,47)(25,38)(26,37)(30,42,40,46)(33,45,34,43)(35,44)(41,48);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,20)(18,24)(19,27)(21,34)(22,32)(23,36)(25,40)(26,39)(28,31)(29,45)(30,41)(33,35)(37,47)(38,46)(42,48)(43,44);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,28)(23,37)(27,31)(29,44)(36,47)(38,48)(42,46)(43,45)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,f^b=fi,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, 145623);