#265094

order	512
nilpotency class	3
isoclinism	isoclinic class of #265019
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	C₂×#3303	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18,24,28)(19,29,32,43)(20,26,36,42)(21,22,27,23)(25,37,40,31)(30,41,47,45)(33,39,46,48)(34,35,44,38);(2,7)(3,12)(5,6)(8,11)(13,14)(15,16)(17,19,23,37,24,32,22,31)(18,25,27,43,28,40,21,29)(20,33,38,47,36,46,35,30)(26,41,44,48,42,45,34,39);(1,3,4,13)(2,8,7,15)(5,12,6,14)(9,16,10,11)(17,20)(18,26)(19,30)(21,34)(22,35)(23,38)(24,36)(25,39)(27,44)(28,42)(29,45)(31,46)(32,47)(33,37)(40,48)(41,43);(2,7)(3,13)(9,10)(12,14)(17,21,22,28,24,27,23,18)(19,29,31,40,32,43,37,25)(20,34,35,42,36,44,38,26)(30,45,46,48,47,41,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,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=ghi,c^b=cg,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172290);