#263789

order	512
nilpotency class	3
isoclinism	isoclinic class of #263779
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#121	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

757 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170985);