#299298

order	512
nilpotency class	3
isoclinism	isoclinic class of #299253
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	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

981 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,24,28)(19,21,31,32)(20,22,27,23)(25,30,29,26);(2,6)(7,14)(8,13)(9,15)(17,19,23,30,24,31,22,26)(18,25,27,32,28,29,20,21);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20,23,18,24,27,22,28)(19,29,30,21,31,25,26,32);(2,6)(7,14)(8,13)(9,15)(17,21,22,29,24,32,23,25)(18,26,20,31,28,30,27,19)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206494);