#115228

order	512
nilpotency class	3
isoclinism	isoclinic class of #115222
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3006	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 22424);