#122591

order	512
nilpotency class	4
isoclinism	isoclinic class of #122544
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29787);