#122885

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

Polycyclic presentation:

2^9|a^2=e,c^2=hi,d^2=gi,e^2=fgi,f^2=h,g^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fh,g^a=gh,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, 30081);