#149020

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

623 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4)(5,6,11,17,12,16,28,25)(7,18,23,30,24,20,35,36)(8,19,26,9,14,10,15,27)(13,29,31,21,32,22,33,34);(5,7,19,29)(6,13,26,36)(8,18,17,33)(9,21,28,35)(10,22,12,24)(11,23,27,34)(14,20,25,31)(15,30,16,32);(5,8,9,16)(6,12,14,27)(7,20,21,32)(10,25,28,15)(11,26,19,17)(13,24,18,34)(22,31,35,36)(23,30,29,33)

Polycyclic presentation:

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

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, 56216);