#2110314

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2292 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)(5,6,10,11)(7,12,8,9)(13,14)(15,19)(16,18)(17,20)(23,27)(26,28);(6,11)(7,8)(13,15,18,17)(14,19,16,20)(22,25)(26,28);(5,7,10,8)(6,12,11,9)(13,15,18,17)(14,20,16,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,16,18,14)(15,19,17,20)(21,23)(22,26)(24,27)(25,28);(5,9,10,12)(6,7,11,8)(13,17,18,15)(14,19,16,20)

Polycyclic presentation:

2^9|a^2=fghi,b^2=g,c^2=fg,d^2=fg,e^2=fg,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, 2017510);