#128196

order	512
nilpotency class	4
isoclinism	isoclinic class of #128055
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

551 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)(3,11,13,7)(4,8)(5,10)(6,9)(12,16,14,15)(17,18,21,26,22,27,35,43)(19,28,30,44,31,37,45,47)(20,25,33,42,34,24,40,41)(23,36,38,32,39,46,48,29);(1,3,5,14,6,13,4,12)(2,7,10,16,9,11,8,15)(17,19)(18,23)(20,32,34,29)(21,30)(22,31)(24,28,25,37)(26,38)(27,39)(33,46,40,36)(35,45)(41,44,42,47)(43,48);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20)(18,24)(19,29)(21,33)(22,34)(23,37)(25,27)(26,41)(28,39)(30,36)(31,32)(35,40)(38,47)(42,43)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fh,c^b=ch,d^b=dh,e^b=eg,h^b=hi,f^c=fi,f^d=fi

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, 35392);