#5955904

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 100663296
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4510 conjugacy classes of subgroups; only the 954 normal subgroups are shown.

normal subgroups only

This group has 4510 conjugacy classes of subgroups — too many to draw. Only its 954 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 — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5863100);