#321778

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

723 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 723 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi

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, 228974);