#6465344

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 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)(4,6)(5,8)(18,23)(19,27)(20,21)(25,26)(28,29)(30,32);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,26,22,25,20,23)(19,24,27,32,28,31,29,30);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,19,20,29,22,28,21,27)(18,24,23,30,25,31,26,32);(9,11,12,16)(10,14,13,15)(19,28)(24,31)(27,29)(30,32);(1,4,5,3)(2,6,8,7)(18,25)(23,26)(24,31)(30,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6372540);