#299714

order	512
nilpotency class	3
isoclinism	isoclinic class of #299653
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	#25622	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fh,d^2=ehi,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,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206910);