#220238

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#3061	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,d^b=dei,e^b=eh,d^c=di,f^c=fh,g^c=gh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127434);