#2606500

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

4122 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333819, #72333840, #72333941
maximal normal subgroup of	#72333819, #72333840, #72333941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2513696);