#213909

order	512
nilpotency class	3
isoclinism	isoclinic class of #213852
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1340 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 190 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,d^c=di,g^c=gi,g^d=gh

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, 121105);