#190342

order	512
nilpotency class	3
isoclinism	isoclinic class of #190230
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 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)(3,7,4,12)(5,10)(6,9)(8,13,16,11)(14,15)(17,18,22,26,23,20,21,25)(19,24,28,31,29,27,30,32);(1,3,5,4)(2,7,10,12)(6,11,14,13)(8,15,16,9)(17,19)(18,24)(20,27)(21,30)(22,28)(23,29)(25,32)(26,31);(1,4,5,3)(2,8)(6,13,14,11)(7,15)(9,12)(10,16)(17,20)(18,23)(19,27)(21,26)(22,25)(24,29)(28,32)(30,31);(17,21,23,22)(18,25,20,26)(19,28,29,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,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, 97538);