#520916

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2142 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428112);