#193565

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6,9,11,8,7,10,12)(13,14)(15,25,27,35)(16,30,33,38)(17,23)(18,24)(19,22)(20,28,36,29)(21,32,37,31)(26,40)(34,41)(39,42)(43,44);(1,2,3,4)(5,7,9,12,8,6,10,11)(13,15,17,29,19,27,18,28)(14,20,23,35,22,36,24,25)(16,26,32,42,33,41,31,43)(21,34,30,44,37,40,38,39);(6,7)(11,12)(13,16,19,33)(14,21,22,37)(15,26,27,41)(17,32,18,31)(20,34,36,40)(23,30,24,38)(25,39,35,44)(28,43,29,42);(14,22)(15,27)(21,37)(23,24)(26,41)(28,29)(30,38)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100761);