#93740

order	512
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1831	C₂²
socle	C₂²	#1325
prime radical	C₂³	#324
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1831 ◁ G
LECS	G ▷ #1831 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

125 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 125 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,4,9,6,11,3,12)(5,8,13,7,14,10,15,16)(17,18,20,25,21,26,31,39,22,24,30,36,32,37,43,46)(19,23,28,33,29,35,42,44,27,34,40,38,41,45,48,47);(1,3,5,13,6,4,14,15)(2,7,8,9,11,16,10,12)(17,19,22,27)(18,23)(20,28,30,40)(21,29,32,41)(24,34)(25,38)(26,35)(31,42,43,48)(33,36)(37,45)(39,47)(44,46)

Polycyclic presentation:

2^9|a^2=d,b^2=eg,c^2=gh,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=chi,d^b=deghi,d^c=dgi,e^d=ei

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, 936);