#115317

order	512
nilpotency class	3
isoclinism	isoclinic class of #115302
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=dh,d^c=di,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22513);