#4245062

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1959 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=ghi,c^2=gh,d^2=gi,e^2=g,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, 4152258);