#428398

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
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

1070 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)(3,12,8,4)(5,17,19,31)(6,10)(7,11)(9,20,23,32)(13,27)(14,26,22,16)(15,28)(18,29)(21,24)(25,30);(1,3)(2,4)(5,18)(6,8)(7,14)(9,15)(10,12)(11,16)(13,19)(17,29)(20,30)(21,22)(23,25)(24,26)(27,31)(28,32);(1,4)(2,8)(3,10)(5,15)(6,12)(7,16)(9,18)(11,22)(13,23)(14,24)(17,30)(19,25)(20,29)(21,26)(27,32)(28,31);(1,5)(2,9,10,23)(3,13,8,18)(4,15)(6,19)(7,20)(11,17,24,31)(12,25)(14,28,22,30)(16,29)(21,32)(26,27)

Polycyclic presentation:

2^9|a^2=ef,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,e^d=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, 335594);