#112820

order	512
nilpotency class	3
isoclinism	isoclinic class of #100117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

956 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,f^c=fh,g^c=ghi

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, 20016);