#347360

order	512
nilpotency class	3
isoclinism	isoclinic class of #347322
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1939 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,e^d=eh,g^d=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254556);