#1597557

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2140 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=g,e^2=fi,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, 1504753);