#352374

order	512
nilpotency class	3
isoclinism	isoclinic class of #352368
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29818	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ci,d^b=di,g^d=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259570);