#411079

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15103
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

780 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)(5,11)(6,9)(7,12)(8,14)(10,20)(15,22)(16,24)(17,23)(19,26)(21,25)(27,29)(28,31)(30,32);(1,3,7,17)(2,8)(4,15,21,30)(5,14,6,16)(9,13)(10,22)(11,23)(12,24)(18,27)(19,29,20,28)(25,31)(26,32);(2,9,12,11)(3,14,17,16)(10,18,26,25)(15,28,30,29);(1,4,6,19,7,21,5,20)(2,10,11,25,12,26,9,18)(3,15,16,29,17,30,14,28)(8,22,23,31,24,32,13,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318275);