#93961

order	512
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	#1729	C₂²
prime radical	C₂⁴	#104
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

239 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 40 points:

(2,5,6,7)(9,10,13,19,15,20,26,40,16,22,25,38,14,21,11,23)(12,27,28,34,30,39,24,17,31,33,32,35,29,18,37,36);(1,2)(3,7)(4,6)(5,8)(9,11,14,13,16,26,15,25)(10,17,21,35,22,36,20,34)(12,28,29,24,31,32,30,37)(18,38,33,23,39,19,27,40)

Polycyclic presentation:

2^9|a^2=d,b^2=eh,c^2=fh,d^2=g,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defh,f^b=fi,g^b=gi,d^c=dhi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1157);