#219527

order	512
nilpotency class	3
isoclinism	isoclinic class of #219525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cei,f^b=fh,d^c=di,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, 126723);