#124818

order	512
nilpotency class	4
isoclinism	isoclinic class of #124728
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1502
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7625
prime radical	C₂³	#421
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

257 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,11,7,13,23,27,8,14,22,10,6,12,4,21)(3,15,17,26,18,28,32,29,20,30,31,9,19,24,16,25);(1,3)(2,9,14,26)(4,16)(5,17)(6,19)(7,18)(8,20)(10,24,11,28)(12,29,13,25)(15,27,30,21)(22,31)(23,32);(1,4,7,5,8,23,6,22)(2,10,12,27,14,11,13,21)(3,16,18,17,20,32,19,31)(9,24,29,30,26,28,25,15)

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=fgh,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,g^b=gi,e^c=ei,e^d=ei

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, 32014);