#3548595

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 2097152
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

2324 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

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, 3455791);