#428404

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15383
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1077 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:

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

Polycyclic presentation:

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

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, 335600);