#254559

order	512
nilpotency class	3
isoclinism	isoclinic class of #254539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#29620	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

869 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)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,21,23)(19,24,29,35)(20,25,30,36)(22,33,32,26)(27,37,34,39)(28,42,43,38)(31,45,44,40)(41,48,46,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(18,23)(19,27,29,34)(24,37,35,39)(25,36)(26,33)(28,41,43,46)(38,47,42,48)(40,45);(4,11)(6,12)(7,14)(9,15)(17,19)(18,24)(20,28)(21,29)(22,34)(23,35)(25,38)(26,37)(27,32)(30,43)(31,46)(33,39)(36,42)(40,47)(41,44)(45,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20)(18,25)(19,28)(21,30)(22,31)(23,36)(24,38)(26,40)(27,41)(29,43)(32,44)(33,45)(34,46)(35,42)(37,47)(39,48)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=i,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,g^c=gh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161755);