#196153

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3180	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1114 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103349);