#149274

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
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

733 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 138 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)(6,11)(7,15)(9,16)(12,25)(13,29)(14,24)(17,30)(18,31)(19,33)(20,32)(22,27)(23,26)(28,35)(34,36);(5,6,8,11)(7,12,17,22)(9,13,20,23)(10,14,21,24)(15,27,30,25)(16,26,32,29)(18,28,19,34)(31,36,33,35);(5,7,16,31,10,18,9,15)(6,12,26,36,14,28,13,27)(8,19,32,30,21,17,20,33)(11,22,29,35,24,34,23,25)

Polycyclic presentation:

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

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, 56470);