#522397

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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

2582 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,17)(18,20)(22,25)(23,24);(1,3,4,2)(6,9)(10,11)(14,17)(18,19)(21,22,26,25)(23,28,24,27);(13,14)(15,18)(16,17)(19,20)(21,23,26,24)(22,27,25,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24,26,23)(22,28,25,27);(5,7)(6,10)(8,12)(9,11)(13,15,16,20)(14,18,17,19)(21,25,26,22)(23,28,24,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=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, 429593);