#267944

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	#60565	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

863 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)(17,18,21,27,23,24,22,28)(19,29,33,37,32,40,30,25)(20,26,34,41,35,38,36,42)(31,45,44,48,46,43,47,39);(18,24)(19,30)(21,22)(25,29)(26,38)(31,44)(32,33)(34,36)(37,40)(39,43)(45,48)(46,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,25)(20,31)(21,30)(22,33)(23,32)(24,37)(26,39)(27,40)(28,29)(34,47)(35,46)(36,44)(38,48)(41,43)(42,45);(1,4,5,6)(2,8,10,9)(3,12,14,13)(7,11,15,16)(17,20)(18,26)(19,31)(21,36)(22,34)(23,35)(24,38)(25,39)(27,42)(28,41)(29,43)(30,44)(32,46)(33,47)(37,48)(40,45)

Polycyclic presentation:

2^9|a^2=fhi,d^2=hi,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, 175140);