#4414516

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
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

3397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4321712);