#3547467

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
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

2391 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=fi,d^2=fi,e^2=f,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, 3454663);