#651619

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

2120 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)(5,6,10,11)(7,12,8,9)(13,14,18,17)(15,20,19,16)(21,22)(23,24)(25,27)(26,28);(6,11)(9,12)(21,23,25,28)(22,24,27,26);(5,7,10,8)(6,12,11,9)(13,15)(14,16)(17,20)(18,19)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(13,16,18,20)(14,19,17,15);(5,9)(6,7)(8,11)(10,12)(13,17,18,14)(15,20,19,16)(21,24,25,26)(22,23,27,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=f,d^2=fgh,e^2=g,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, 558815);