#93028

order	512
nilpotency class	4
isoclinism	isoclinic class of #93021
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1289
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×C₈×Q₈	C₂²
prime radical	C₂³	#340
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

191 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fghi,d^2=i,e^2=fgi,f^2=hi,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gi,c^b=ch,d^b=dg,d^c=di

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, 224);