#264994

order	512
nilpotency class	3
isoclinism	isoclinic class of #264939
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

818 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)(3,10)(4,8)(5,7)(6,12)(9,14)(11,16)(13,15)(17,18,24,30)(19,31,34,27)(20,35,38,44)(21,25,33,41)(22,29,23,28)(26,39,43,37)(32,46,47,40)(36,45,48,42);(1,3,4,12)(2,6,8,10)(5,11,14,13)(7,15,9,16)(17,19,23,33,24,34,22,21)(18,25,29,27,30,41,28,31)(20,32,39,48,38,47,37,36)(26,40,44,42,43,46,35,45);(2,7)(3,11)(5,14)(8,9)(12,13)(15,16)(17,20)(18,26)(19,32)(21,36)(22,37)(23,39)(24,38)(25,40)(27,42)(28,35)(29,44)(30,43)(31,45)(33,48)(34,47)(41,46);(2,8)(6,10)(7,9)(15,16)(17,21,22,34,24,33,23,19)(18,27,28,25,30,31,29,41)(20,36,37,47,38,48,39,32)(26,42,35,40,43,45,44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172190);