#536332

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2066 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=gh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 443528);