#267918

order	512
nilpotency class	3
isoclinism	isoclinic class of #267899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,f^c=fi,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, 175114);