#121069

order	512
nilpotency class	3
isoclinism	isoclinic class of #10690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#401
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

994 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28265);