#654432

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
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

2387 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)(7,12,11,8)(13,14,17,15)(16,18,20,19)(21,22,25,24)(23,26,27,28);(14,15)(16,20)(23,27)(26,28);(5,7)(6,8)(9,12)(10,11)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,8,10,12)(6,11,9,7);(5,9,10,6)(7,12,11,8)(13,16)(14,18)(15,19)(17,20)(21,24,25,22)(23,28,27,26)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 561628);