#1541896

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2856 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1449092);