#264000

order	512
nilpotency class	3
isoclinism	isoclinic class of #263983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3558	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 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)(3,8)(4,14)(5,10)(6,11)(7,12)(9,15)(13,16)(17,18,23,27)(19,28,33,30)(20,34,38,25)(21,40,26,22)(24,32,43,31)(29,44,45,46)(35,42,47,48)(36,39,41,37);(1,3)(2,8)(4,12)(5,13)(6,9)(7,14)(10,16)(11,15)(17,19,22,31,23,33,40,32)(18,24,26,28,27,43,21,30)(20,35,37,45,38,47,39,29)(25,44,36,48,34,46,41,42);(1,4,7,8)(2,9,12,13)(3,11,14,10)(5,15,6,16)(17,20)(18,25)(19,29)(21,41)(22,37)(23,38)(24,42)(26,36)(27,34)(28,44)(30,46)(31,35)(32,47)(33,45)(39,40)(43,48);(1,5,7,6)(2,10,12,11)(3,13,14,9)(4,15,8,16)(17,21,22,18,23,26,40,27)(19,30,31,24,33,28,32,43)(20,36,37,34,38,41,39,25)(29,44,35,48,45,46,47,42)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=ghi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171196);