#299853

order	512
nilpotency class	3
isoclinism	isoclinic class of #299805
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	#62342	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

905 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)(18,24)(19,20)(21,22)(25,26)(27,28)(30,31);(2,6)(3,10)(5,12)(7,14)(17,18,22,27,23,28,21,24)(19,29,20,26,31,32,30,25);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,19,22,20,23,31,21,30)(18,25,24,32,28,26,27,29);(2,6)(3,10)(8,13)(11,16)(17,20,22,31,23,30,21,19)(18,26,27,32,28,25,24,29)

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=ei,e^2=h,f^2=i,g^2=h,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^b=gh,d^c=di,g^c=ghi

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, 207049);