#122508

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	#3169	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

270 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 86 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)(17,18,20,25,22,26,33,40,21,24,32,37,34,38,45,47)(19,27,29,39,30,41,43,48,28,23,31,35,42,36,44,46);(2,7)(3,5)(4,12)(8,9)(10,16)(11,13)(14,15)(17,19)(18,23)(20,31)(21,28)(22,30)(24,27)(25,39)(26,36)(29,32)(33,44)(34,42)(35,37)(38,41)(40,48)(43,45)(46,47);(1,3,6,5)(2,8,10,14)(4,11,13,12)(7,15,16,9)(18,24)(19,28)(25,37)(26,38)(29,31)(30,42)(40,47)(43,44)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=fhi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=chi,d^b=dfi,e^b=ei,f^b=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29704);