#232565

order	512
nilpotency class	3
isoclinism	isoclinic class of #232560
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60805	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=ei,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, 139761);