#300364

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	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

907 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fh,d^2=eh,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, 207560);