#195642

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1262 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1262 conjugacy classes of subgroups — too many to draw. Only its 282 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,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, 102838);