#442178

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15702
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

947 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349374);