#249170

order	512
nilpotency class	3
isoclinism	isoclinic class of #249139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1228 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1228 conjugacy classes of subgroups — too many to draw. Only its 188 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156366);