#451059

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15984
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1269 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358255);