#268695

order	512
nilpotency class	3
isoclinism	isoclinic class of #268627
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

763 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=h,e^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi

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, 175891);