#4599098

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×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

3199 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4506294);