#245467

order	512
nilpotency class	3
isoclinism	isoclinic class of #245447
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152663);