#97881

order	512
nilpotency class	3
isoclinism	isoclinic class of #97164
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2682	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

561 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5077);