#128931

order	512
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,f^2=ghi,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36127);