#247016

order	512
nilpotency class	3
isoclinism	isoclinic class of #246987
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	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1044 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,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, 154212);