#6495599

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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	#35278	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1633 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:

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

Polycyclic presentation:

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

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, 6402795);