#4125355

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2040 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=f,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4032551);