#531925

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2150 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)(5,6,9,10)(7,11,8,12)(14,16)(15,17)(22,26)(23,28);(6,10)(11,12)(13,14,18,16)(15,20,17,19);(5,7,9,8)(6,11,10,12)(13,15,18,17)(14,19,16,20)(21,22,25,26)(23,27,28,24);(5,8,9,7)(6,12,10,11)(13,16,18,14)(15,20,17,19)(21,23)(22,27)(24,26)(25,28);(5,6)(7,11)(8,12)(9,10)(13,17,18,15)(14,20,16,19)(21,24,25,27)(22,23,26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef

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, 439121);