#190325

order	512
nilpotency class	3
isoclinism	isoclinic class of #190218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

924 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)(3,8,4,15)(5,10)(6,12)(7,11)(9,13,16,14)(17,18,21,25,22,24,20,29)(19,26,27,23,28,32,30,31);(1,3,6,4)(2,8,12,15)(5,13,7,14)(9,10,16,11);(1,4,6,3)(2,9)(5,14,7,13)(8,10)(11,15)(12,16)(17,19)(18,23)(20,30)(21,27)(22,28)(24,31)(25,32)(26,29);(1,5,6,7)(2,10,12,11)(3,13,4,14)(8,16,15,9)(18,24)(23,31)(25,29)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^b=gh,f^c=fi,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, 97521);