#250109

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
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	#61205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1142 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,g^c=gi,e^d=eh,f^d=fi

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, 157305);