#237606

order	512
nilpotency class	3
isoclinism	isoclinic class of #237595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1288 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,21,24)(19,25,26,32)(20,28,30,23)(22,29,31,27);(1,3)(2,8)(4,7)(5,13)(6,14)(9,10)(11,16)(12,15);(1,4,7,3)(2,9,11,12)(5,14,6,13)(8,10,16,15)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(19,26)(22,31)(25,32)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144802);