#233173

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1278 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1278 conjugacy classes of subgroups — too many to draw. Only its 274 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=eh,f^c=fi,g^c=gh

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, 140369);