#3038382

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2449 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)(7,8)(9,11)(10,12)(14,15)(16,20)(22,25)(26,28);(1,2,3,4)(5,7,9,12)(6,10,11,8)(14,15)(18,19)(23,27)(26,28);(1,2)(3,4)(13,14,17,15)(16,18,20,19)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,12)(9,11)(13,15,17,14)(16,19,20,18)(21,23,24,27)(22,26,25,28);(5,8,9,10)(6,7,11,12)(13,16,17,20)(14,18,15,19)(21,22)(23,26)(24,25)(27,28)

Polycyclic presentation:

2^9|b^2=fhi,c^2=i,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,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2945578);