#561746

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2986 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 468942);