#93701

order	512
nilpotency class	5
isoclinism	isoclinic class of #93701
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂³×C₄	C₂×C₈
Frattini, cosocle	#1779	C₂²
prime radical	C₂²×#121	C₄
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

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

normal subgroups only

This group has 409 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,c^2=h,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ch,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, 897);