#235328

order	512
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60558	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142524);