#119916

order	512
nilpotency class	3
isoclinism	isoclinic class of #119915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,23,20,28)(21,26)(22,25)(24,29,32,27)(30,31);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,21,20)(18,23,26,28)(22,29,30,27)(24,25,32,31);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(17,20,21,19)(18,24)(22,29,30,27)(23,31)(25,28)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,g^b=gh,e^c=ehi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27112);