#845607

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

4177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(2,3)(6,8)(13,14)(15,16)(17,19)(18,20);(1,2,4,3)(10,12)(13,15,17,20)(14,16,19,18);(9,10)(11,12)(13,15)(14,18)(16,19)(17,20);(1,2,4,3)(5,6)(7,8)(13,15)(14,18)(16,19)(17,20);(1,3,4,2)(13,16,17,18)(14,15,19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327774, #72327799, #72327882
maximal normal subgroup of	#72327774, #72327799, #72327882

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752803);