#300366

order	512
nilpotency class	3
isoclinism	isoclinic class of #300333
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

965 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:

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

Polycyclic presentation:

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

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, 207562);