#651989

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

2446 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)(9,10)(11,12)(13,14)(15,17)(16,19)(18,20)(21,22,26,25)(23,28,24,27);(6,10)(8,11)(13,15,18,16)(14,19,20,17)(21,23,26,24)(22,27,25,28);(5,7)(6,11)(8,10)(9,12)(13,14)(15,17)(16,19)(18,20)(21,24,26,23)(22,27,25,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,15)(14,19,20,17);(5,8,9,11)(6,7,10,12)(13,17,18,19)(14,15,20,16)(21,25,26,22)(23,27,24,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fgh,d^2=h,e^2=f,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, 559185);