#331065

order	512
nilpotency class	3
isoclinism	isoclinic class of #331055
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238261);