#4131642

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2670	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

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, 4038838);