#238420

order	512
nilpotency class	3
isoclinism	isoclinic class of #238414
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

755 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)(5,13)(6,9)(7,11)(12,16)(14,15)(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)(8,13)(11,14)(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)(2,7)(4,11)(5,14)(6,12)(8,16)(9,15)(10,13)(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)(2,8,9,13)(3,11,12,14)(7,16,15,10)(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,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,e^c=eh,f^c=fhi

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, 145616);