#128924

order	512
nilpotency class	4
isoclinism	isoclinic class of #11994
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,f^c=fi

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, 36120);