#193674

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fh,g^b=gh,d^c=di,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, 100870);