#116070

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#397
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23266);