#265174

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 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)(4,10)(5,9)(6,8)(7,14)(12,16)(13,15)(17,18,24,30)(19,27,33,42)(20,34,37,46)(21,38,32,25)(22,29,23,28)(26,40,44,36)(31,47,48,41)(35,45,39,43);(1,3)(2,7)(4,13)(5,14)(6,12)(8,15)(9,11)(10,16)(17,19,23,21,24,33,22,32)(18,25,29,42,30,38,28,27)(20,31,40,39,37,48,36,35)(26,41,46,43,44,47,34,45);(2,8)(3,12)(4,6)(9,10)(13,14)(15,16)(17,20)(18,26)(19,31)(21,39)(22,36)(23,40)(24,37)(25,41)(27,45)(28,34)(29,46)(30,44)(32,35)(33,48)(38,47)(42,43);(1,4,5,6)(2,8,9,10)(3,13,14,12)(7,15,11,16)(17,21,23,33,24,32,22,19)(18,27,29,25,30,42,28,38)(20,35,40,31,37,39,36,48)(26,43,46,47,44,45,34,41)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=eh,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,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 172370);