#124336

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1530
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#376	C₂³
socle	C₂	#7578
prime radical	#3455	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 419 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) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31532);