#300378

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	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

906 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ehi,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, 207574);