#139199

order	512
nilpotency class	4
isoclinism	isoclinic class of #139160
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46395);