#219497

order	512
nilpotency class	3
isoclinism	isoclinic class of #219491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

592 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)(17,18,23,27,24,29,22,28)(19,30,32,35,33,41,25,20)(21,36,39,45,40,44,38,26)(31,37,43,48,47,34,42,46);(1,3,6,9,7,2,5,10)(4,11,12,16,13,8,14,15)(17,19)(18,20,29,35)(21,31)(22,25)(23,32)(24,33)(26,34,45,37)(27,30,28,41)(36,46,44,48)(38,42)(39,43)(40,47);(1,4)(2,8)(3,11)(5,14)(6,12)(7,13)(9,16)(10,15)(17,20)(18,25,29,32)(19,27,33,28)(21,37)(22,30)(23,41)(24,35)(26,43,45,42)(31,44,47,36)(34,40)(38,48)(39,46);(1,5,7,6)(2,9,3,10)(4,12,13,14)(8,15,11,16)(17,21,22,38,24,40,23,39)(18,26,27,36,29,45,28,44)(19,31,25,42,33,47,32,43)(20,34,30,46,35,37,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,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=ce,f^b=fh,d^c=di,g^c=gh

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, 126693);