#193909

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101105);