#242642

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

766 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149838);