#1434169

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1341365);