#2281116

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

3359 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fghi,c^2=fgi,d^2=f,e^2=f,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, 2188312);