#5956480

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 67108864
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3106 conjugacy classes of subgroups; only the 722 normal subgroups are shown.

normal subgroups only

This group has 3106 conjugacy classes of subgroups — too many to draw. Only its 722 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) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=gi,e^2=g,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, 5863676);