#434414

order	512
nilpotency class	3
isoclinism	isoclinic class of #420416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15501
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1146 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=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, 341610);