#636020

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

6037 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72330836, #72330845, #72330935
maximal normal subgroup of	#72330836, #72330845, #72330935
maximal quotient of	#72333167

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 543216);