#456509

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16283
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1140 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:

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

Polycyclic presentation:

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

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, 363705);