#115448

order	512
nilpotency class	3
isoclinism	isoclinic class of #115437
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

810 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)(3,10)(4,6)(5,15)(7,9)(8,11)(12,16)(13,14)(17,18,22,27,23,28,21,26)(19,24,30,38,31,39,36,42)(20,32,35,25,33,41,34,40)(29,43,45,48,44,47,46,37);(1,3,6,12)(2,8,4,14)(5,13,7,11)(9,16,15,10)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,37)(26,42)(27,38)(28,39)(32,47)(33,44)(34,46)(35,45)(40,48)(41,43);(1,4,6,2)(3,10,12,16)(5,15,7,9)(8,11,14,13)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,32)(28,40)(30,46)(31,44)(36,45)(38,47)(39,48)(42,43)

Polycyclic presentation:

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

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, 22644);