#213968

order	512
nilpotency class	3
isoclinism	isoclinic class of #213852
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2403	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1019 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1019 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,g^2=i,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, 121164);