#267942

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	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

844 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,d^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, 175138);