#93010

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×C₈×Q₈	C₂²
socle	C₂²	#1282
prime radical	C₂³	#336
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

193 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=gh,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dfhi,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206);