#4606590

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2869 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fg,d^2=fg,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, 4513786);