#193705

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
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

1138 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)(5,6,10,13,11,7,9,14)(8,16,18,12,19,20,17,15)(21,22,26,29,25,28,34,36)(23,24,31,33,30,32,27,35);(1,2,3,4)(5,7,11,6)(8,15,19,12)(9,14,10,13)(16,18,20,17)(21,23,26,31,25,30,34,27)(22,24,29,33,28,32,36,35);(5,8,11,19)(6,12,7,15)(9,18,10,17)(13,16,14,20)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,36)(34,35);(5,9,11,10)(6,13,7,14)(8,17,19,18)(12,20,15,16)(23,30)(24,32)(27,31)(33,35)

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=dh,d^b=dhi,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

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100901);