#577219

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2894 conjugacy classes of subgroups; only the 846 normal subgroups are shown.

normal subgroups only

This group has 2894 conjugacy classes of subgroups — too many to draw. Only its 846 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fghi,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

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, 484415);