#536148

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=f,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, 443344);