#6532608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439804);