#453921

order	512
nilpotency class	3
isoclinism	isoclinic class of #453920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16224
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

666 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361117);