#6464967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(9,10)(11,14)(12,13)(15,16)(17,18,23,24,22,19,21,26)(20,25,28,31,30,27,29,32);(2,5)(6,8)(18,24)(19,26)(20,28)(21,23)(25,27)(29,30);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19,21,24,22,18,23,26)(20,25,29,32,30,27,28,31);(17,20,23,28,22,30,21,29)(18,25,24,31,19,27,26,32);(9,11,12,15)(10,14,13,16)(18,19)(24,26)(25,27)(31,32)

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=h,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=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, 6372163);