#6469738

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	#34624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376934);