#355100

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2000 conjugacy classes of subgroups; only the 394 normal subgroups are shown.

normal subgroups only

This group has 2000 conjugacy classes of subgroups — too many to draw. Only its 394 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) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262296);