#115170

order	512
nilpotency class	3
isoclinism	isoclinic class of #115163
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, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1086 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,g^b=gh,d^c=di,g^c=gh

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, 22366);