#2138977

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

2305 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)(6,9)(7,11)(13,14)(15,19)(16,18)(17,20)(21,22,25,26)(23,24,28,27);(6,9)(10,12)(13,15,18,17)(14,19,16,20)(22,26)(23,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,20,16,19)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,17,20);(5,7)(6,10)(8,11)(9,12)(13,17,18,15)(14,19,16,20)(21,24,25,27)(22,23,26,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=fgh,d^2=gh,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, 2046173);