#6342404

order	512
nilpotency class	2
isoclinism	isoclinic class of #6342371
automorphism	Aut(G) order 134217728
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2722 conjugacy classes of subgroups; only the 978 normal subgroups are shown.

normal subgroups only

This group has 2722 conjugacy classes of subgroups — too many to draw. Only its 978 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) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249600);