#1259296

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#546	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(2,4)(6,8)(13,14,17,16)(15,19,20,18);(9,10,11,12)(13,15)(14,18)(16,19)(17,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,17,16)(15,19,20,18);(5,6)(7,8)(13,15)(14,19)(16,18)(17,20);(1,2)(3,4)(5,6)(7,8)(13,16,17,14)(15,19,20,18)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280828, #72281089, #72281166
maximal normal subgroup of	#72280828, #72281089, #72281166

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1166492);