#119738

order	512
nilpotency class	3
isoclinism	isoclinic class of #119733
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

540 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,e^c=ei,g^c=gh

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, 26934);