#127661

order	512
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=g,e^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=cgh,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, 34857);