#97039

order	512
nilpotency class	3
isoclinism	isoclinic class of #97030
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

686 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,26,28,23)(20,22)(21,25)(24,29)(27,31,32,30)(33,34,36,40,37,39,45,47)(35,41,43,48,42,38,44,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,22)(20,29)(23,30)(24,25)(26,31)(27,32)(33,35)(34,38)(36,44)(37,42)(39,41)(40,48)(43,45)(46,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,28)(18,23,25,26)(20,27,29,32)(22,30,24,31)(35,42)(38,41)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^b=fgh,f^c=fh

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, 4235);