#515681

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂×#2430	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316041
maximal normal subgroup of	#72316041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422877);