#299582

order	512
nilpotency class	3
isoclinism	isoclinic class of #299541
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	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

927 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)(3,11,4,14)(5,16,12,10)(7,15,9,13)(17,18)(19,24)(20,27)(21,25)(22,31)(23,26)(28,29)(30,32);(1,3,5,13,6,4,12,15)(2,7,10,11,8,9,16,14)(17,19)(18,24)(20,25)(21,27)(22,32)(23,28)(26,29)(30,31);(2,8)(7,9)(10,16)(11,14)(17,20,22,24,23,30,21,29)(18,19,25,27,26,28,31,32);(1,4,5,15,6,3,12,13)(2,9,10,14,8,7,16,11)(17,21,23,22)(18,25,26,31)(19,27,28,32)(20,29,30,24)

Polycyclic presentation:

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

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, 206778);