#423178

order	512
nilpotency class	3
isoclinism	isoclinic class of #423178
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15312
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1327 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:

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

Polycyclic presentation:

2^9|c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 330374);