#267920

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

853 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)(17,18,23,36,22,24,21,27)(19,28,35,25,31,39,29,37)(20,26,33,42,34,38,32,41)(30,45,44,40,46,43,47,48);(18,24)(19,29)(21,23)(25,39)(26,38)(28,37)(30,44)(31,35)(32,33)(40,45)(43,48)(46,47);(1,3,5,11)(2,6,7,14)(4,12,13,10)(8,16,15,9)(17,19,22,31)(18,25,24,37)(20,30,34,46)(21,35,23,29)(26,40,38,48)(27,39,36,28)(32,44,33,47)(41,43,42,45);(2,7)(6,14)(8,15)(9,16)(17,20,22,34)(18,26,24,38)(19,30,31,46)(21,33,23,32)(25,40,37,48)(27,42,36,41)(28,43,39,45)(29,44,35,47)

Polycyclic presentation:

2^9|a^2=fh,c^2=i,d^2=h,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, 175116);