#254514

order	512
nilpotency class	3
isoclinism	isoclinic class of #254491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

778 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161710);