#459350

order	512
nilpotency class	3
isoclinism	isoclinic class of #453994
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16339
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

822 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=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, 366546);