#238382

order	512
nilpotency class	3
isoclinism	isoclinic class of #238377
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

767 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)(3,9,11,15)(4,7)(5,8)(6,12,10,13)(14,16)(17,18)(19,23)(20,30,33,24)(21,25)(22,26)(27,36,31,39)(28,37)(29,38)(32,41,34,40)(35,42)(43,46,44,47)(45,48);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,15,13)(14,16)(17,19)(18,23)(20,31,33,27)(21,28)(22,29)(24,39,30,36)(25,37)(26,38)(32,44,34,43)(35,45)(40,47,41,46)(42,48);(1,3)(2,6)(4,13)(5,11)(7,15)(8,10)(9,16)(12,14)(17,20,22,33)(18,24,26,30)(19,27,29,31)(21,34,35,32)(23,36,38,39)(25,41,42,40)(28,44,45,43)(37,47,48,46);(17,21,22,35)(18,25,26,42)(19,28,29,45)(20,32,33,34)(23,37,38,48)(24,40,30,41)(27,43,31,44)(36,46,39,47)

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,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, 145578);