#120099

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

558 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=i,f^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fhi,e^c=ei,f^c=fhi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27295);