#250037

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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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)(17,18,22,20)(19,25,27,32)(21,29,31,24)(23,30,26,28);(2,6)(4,14)(7,10)(8,9)(11,13)(15,16)(17,18,22,20)(19,23,27,26)(21,29,31,24)(25,28,32,30);(1,2,5,9)(3,7,12,15)(4,8,14,6)(10,13,16,11)(17,19,22,27)(18,23,20,26)(21,30,31,28)(24,32,29,25);(1,3)(2,7)(4,13)(5,12)(6,10)(8,16)(9,15)(11,14)(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=i,c^2=hi,d^2=i,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, 157233);