#115791

order	512
nilpotency class	3
isoclinism	isoclinic class of #99835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

740 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,21,30,36)(15,19,25,33,24,32,40,43)(16,26,27,37,28,20,29,35)(23,31,39,42,38,34,41,44);(1,2,3,4)(6,9)(10,11)(13,15,18,24)(14,19,21,32)(16,23,28,38)(17,25,30,40)(20,34,26,31)(22,33,36,43)(27,39,29,41)(35,44,37,42);(6,10)(8,12)(9,11)(13,16,18,28)(14,20,21,26)(15,23,24,38)(17,29,30,27)(19,31,32,34)(22,37,36,35)(25,41,40,39)(33,44,43,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22987);