#102372

order	512
nilpotency class	3
isoclinism	isoclinic class of #102154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

409 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=gh,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,f^b=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9568);