#697989

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3059 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=h,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, 605185);