#317977

order	512
nilpotency class	3
isoclinism	isoclinic class of #317966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#555
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,f^c=fi,g^c=ghi

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, 225173);