#111301

order	512
nilpotency class	3
isoclinism	isoclinic class of #96944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2656	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

940 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=ei

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, 18497);