#237282

order	512
nilpotency class	3
isoclinism	isoclinic class of #237257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1163 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144478);