#5083259

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

3311 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,17)(18,19)(21,22)(23,27)(24,26)(25,28);(5,7,8,12)(6,9,10,11)(13,15)(14,16)(17,18)(19,20)(21,22)(23,27)(24,26)(25,28);(6,10)(9,11)(14,19)(15,17)(21,22,24,26)(23,25,28,27);(1,2,3,4)(5,7)(6,11)(8,12)(9,10)(13,16,18,20)(14,15,19,17)(21,22)(23,25)(24,26)(27,28);(5,6)(7,11)(8,10)(9,12)(13,17,18,15)(14,20,19,16)(21,23,24,28)(22,25,26,27)

Polycyclic presentation:

2^9|b^2=gh,c^2=fg,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 4990455);