#846850

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

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

4381 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332356, #72332360, #72332388
maximal normal subgroup of	#72332356, #72332360, #72332388
maximal quotient of	#72333171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 754046);