#670060

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2250 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)(7,11)(8,12)(9,10)(13,14,18,20)(15,16,17,19)(21,22,24,26)(23,25,27,28);(1,2,3,4)(6,10)(8,11)(13,15,18,17)(14,16,20,19);(1,2)(3,4)(13,16,18,19)(14,15,20,17)(21,22)(23,25)(24,26)(27,28);(5,7,9,12)(6,11,10,8)(13,14)(15,16)(17,19)(18,20);(5,8,9,11)(6,12,10,7)(13,17,18,15)(14,19,20,16)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=f,e^2=fg,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, 577256);