#5955306

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 50331648
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

5678 conjugacy classes of subgroups; only the 1042 normal subgroups are shown.

normal subgroups only

This group has 5678 conjugacy classes of subgroups — too many to draw. Only its 1042 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,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, 5862502);