#156565

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de

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, 63761);