#6489106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1863 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei

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, 6396302);