#423839

order	512
nilpotency class	3
isoclinism	isoclinic class of #418530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15316
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 331035);