#127864

order	512
nilpotency class	4
isoclinism	isoclinic class of #11753
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 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

547 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:

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

Polycyclic presentation:

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

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, 35060);