#120322

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 474 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) = 256

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

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

Polycyclic presentation:

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

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, 27518);