#2329037

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×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

2826 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)(7,9)(8,10)(11,12)(13,14,17,20)(15,19,16,18)(22,26)(24,27);(6,7)(10,11)(13,14,17,20)(15,18,16,19)(21,22,25,26)(23,24,28,27);(8,12)(10,11)(13,15,17,16)(14,18,20,19)(21,23,25,28)(22,27,26,24);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,24)(22,23)(25,27)(26,28);(5,8,9,12)(6,10,7,11)(13,16,17,15)(14,19,20,18)(21,23)(22,24)(25,28)(26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 2236233);