#321823

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 777 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,d^c=dh,f^c=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, 229019);