#248545

order	512
nilpotency class	3
isoclinism	isoclinic class of #248503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61699	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 155741);