#6477058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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	#24382	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,27,29,31)(20,24,23,32)(21,30,28,26);(9,11)(10,13)(12,15)(14,16)(17,19,28,32,22,29,21,24)(18,23,30,31,25,20,26,27);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(19,29)(20,23)(24,32)(27,31);(11,15)(13,16)(17,20)(18,24)(19,30)(21,31)(22,23)(25,32)(26,29)(27,28);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,25)(20,23)(26,30)(27,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6384254);