#474441

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16972
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

529 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=h,c^2=g,d^2=ef,e^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,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^d=gi

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, 381637);