#6342398

order	512
nilpotency class	2
isoclinism	isoclinic class of #6342371
automorphism	Aut(G) order 7340032
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3573 conjugacy classes of subgroups; only the 722 normal subgroups are shown.

normal subgroups only

This group has 3573 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) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fgh,c^2=fg,d^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, 6249594);