#2031995

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,23,24,22)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,3,6,5)(2,7,4,8)(11,16)(13,15)(17,18,21,20)(19,22,24,23)(26,30)(27,34)(28,37)(38,39);(1,2,6,4)(3,8,5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,24,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,4,6,2)(3,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,22,20,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,5,6,3)(2,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,23,24,22)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fghi,c^2=fghi,d^2=fghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1939191);