#122903

order	512
nilpotency class	4
isoclinism	isoclinic class of #122859
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 302 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=gi,e^2=fgh,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,c^b=ch,d^b=dg,e^b=efgh,f^b=fh,g^b=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30099);