#249742

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
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	#61668	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1104 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)(3,10)(4,6)(5,9)(7,11)(8,14)(12,16)(13,15)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(2,6)(4,14)(7,10)(8,9)(11,13)(15,16)(20,29)(23,28)(25,32)(27,30);(1,2,5,9)(3,7,12,15)(4,8,14,6)(10,13,16,11)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,28)(27,29);(1,3,5,12)(2,7,9,15)(4,13,14,11)(6,10,8,16)(20,29)(23,28)(25,32)(27,30)

Polycyclic presentation:

2^9|a^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=de,e^a=eh,f^a=fi,c^b=cg,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156938);