#96996

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
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	#2537	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

558 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,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, 4192);