#102874

order	512
nilpotency class	3
isoclinism	isoclinic class of #102873
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	#23918	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1089 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,d^c=dh,e^c=eg,f^c=fg

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, 10070);