#219514

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
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

570 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ehi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cei,f^b=fh,d^c=di,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126710);