#127113

order	512
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

563 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=hi,b^a=bd,d^a=dg,f^a=fi,g^a=gi,c^b=ch,e^b=egi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34309);