#502378

order	512
nilpotency class	4
isoclinism	isoclinic class of #502377
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25321
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2213	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409574);