#249738

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
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
prime radical	#18569	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=hi,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,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gh,e^d=eh,f^d=fi

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, 156934);