#249775

order	512
nilpotency class	3
isoclinism	isoclinic class of #249759
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	#18566	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1069 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:

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

Polycyclic presentation:

2^9|b^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,d^b=di,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156971);