#93708

order	512
nilpotency class	5
isoclinism	isoclinic class of #93707
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1317
commutator	C₂³×C₄	C₂×C₈
Frattini, cosocle	#1779	C₂²
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1779 ◁ G
LECS	G ▷ #1779 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

226 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=g,c^2=h,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=chi,d^b=deh,f^b=fi,d^c=dg,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, 904);