#427436

order	512
nilpotency class	3
isoclinism	isoclinic class of #427436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15369
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1285 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:

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

Polycyclic presentation:

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

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, 334632);