#331051

order	512
nilpotency class	3
isoclinism	isoclinic class of #331031
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

501 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)(3,9)(4,7)(5,8)(6,11)(10,14)(12,16)(13,15)(17,18,22,25,23,26,21,20)(19,27,28,24,31,29,30,32);(1,3)(2,6)(4,11)(5,10)(7,9)(8,16)(12,13)(14,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,31)(25,27)(26,32);(2,7)(3,10)(5,13)(6,14)(9,16)(11,12)(18,20)(19,28)(21,22)(25,26)(27,29)(30,31);(17,20,21,26,23,25,22,18)(19,29,28,32,31,27,30,24)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238247);