#436272

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15554
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 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)(3,13)(4,18,21,26)(5,11)(6,10)(7,12)(8,17)(9,20,25,19)(14,24)(15,23,30,32)(16,22)(27,29,31,28);(3,14,16,17)(8,22,24,13)(15,29,30,28)(23,27,32,31);(1,3)(2,8)(4,15)(5,14)(6,17)(7,16)(9,23)(10,22)(11,13)(12,24)(18,31)(19,28)(20,29)(21,30)(25,32)(26,27);(1,4,5,19,7,21,6,20)(2,9,10,18,12,25,11,26)(3,15,14,28,16,30,17,29)(8,23,22,31,24,32,13,27)

Polycyclic presentation:

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

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, 343468);