#397746

order	512
nilpotency class	3
isoclinism	isoclinic class of #397746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304942);