#4122678

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
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×#2407	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3100 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)(5,6,10,8)(7,12,11,9)(13,14)(15,16)(17,19)(18,20)(21,22)(23,26)(24,28)(25,27);(5,7,10,11)(6,9,8,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,28)(22,26,27,24);(7,11)(9,12)(14,19)(15,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,8,10,6)(7,9,11,12)(13,16,17,18)(14,20,19,15)(21,24)(22,23)(25,26)(27,28);(5,9)(6,11)(7,8)(10,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,d^2=f,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, 4029874);