#4037234

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3944430);