#321686

order	512
nilpotency class	3
isoclinism	isoclinic class of #278195
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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 780 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) = 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)(11,16)(14,18)(15,20)(21,22,24,28,25,27,33,35)(23,29,30,34,31,36,32,26);(1,2,3,4)(5,7,9,16,10,15,18,8)(6,11,14,13,12,17,19,20);(2,4)(6,12)(7,15)(8,16)(14,19)(21,23,25,31)(22,26,27,34)(24,32,33,30)(28,36,35,29);(5,8,18,15,10,16,9,7)(6,13,19,11,12,20,14,17)(22,27)(26,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=egh,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^c=dh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228882);