#122637

order	512
nilpotency class	4
isoclinism	isoclinic class of #122554
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×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 334 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)(3,11)(4,8)(5,14,15,9)(6,10)(7,12,16,13)(17,18,21,26,23,27,22,38)(19,28,31,42,29,45,32,48)(20,33,35,25,37,43,36,44)(24,34,40,46,39,47,41,30);(1,3,6,4)(2,7,10,16)(5,12,15,13)(8,9,11,14)(17,19)(18,24)(20,34)(21,32)(22,31)(23,29)(25,42)(26,40)(27,39)(28,33)(30,35)(36,46)(37,47)(38,41)(43,45)(44,48);(1,4,6,3)(2,8,10,11)(5,13,15,12)(7,9,16,14)(19,29)(24,39)(28,45)(30,46)(31,32)(34,47)(40,41)(42,48)

Polycyclic presentation:

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

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, 29833);