#93674

order	512
nilpotency class	5
isoclinism	isoclinic class of #7339
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂²×C₄	C₂×C₁₆
Frattini, cosocle	#1817	C₂²
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1817 ◁ G
LECS	G ▷ #1817 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 34 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=i,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ci,d^b=dei,f^b=fi,d^c=dg,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, 870);