#245485

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

1053 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)(3,11)(4,12)(5,10)(6,9)(7,14)(8,16)(13,15)(17,18,23,26)(19,24,28,32)(20,29,27,31)(21,25,22,30);(1,3)(2,4)(5,7)(6,13)(8,15)(9,14)(10,11)(12,16)(17,19,23,28)(18,24,26,32)(20,22,27,21)(25,29,30,31);(1,4)(2,7)(3,8)(5,11)(6,14)(9,12)(10,13)(15,16)(17,20)(18,24)(19,21)(22,28)(23,27)(25,31)(26,32)(29,30);(1,4)(2,3)(5,15)(6,14)(7,8)(9,13)(10,12)(11,16)(17,21,23,22)(18,25,26,30)(19,27,28,20)(24,31,32,29)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,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, 152681);