#688963

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2104 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)(7,8)(9,11)(10,12)(13,14,18,17)(15,20,19,16)(23,27)(26,28);(5,7,9,12)(6,10,11,8)(21,22,25,24)(23,28,27,26);(6,11)(8,10)(13,15)(14,16)(17,20)(18,19)(21,23,25,27)(22,26,24,28);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,16,18,20)(14,19,17,15)(21,22,25,24)(23,26,27,28);(5,7)(6,8)(9,12)(10,11)(13,17,18,14)(15,20,19,16)(21,24,25,22)(23,28,27,26)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=h,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 596159);