#321800

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

691 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)(7,15)(8,18)(9,20)(10,12)(11,17)(13,19)(14,16)(21,22,26,28,25,30,24,29)(23,27,33,34,31,35,32,36);(1,2,3,4)(5,7,9,17,10,8,16,19)(6,11,14,15,12,13,20,18)(21,23)(22,27)(24,32)(25,31)(26,33)(28,34)(29,36)(30,35);(2,4)(6,12)(7,8)(14,20)(17,19)(22,28)(24,26)(27,34)(29,30)(32,33)(35,36);(5,8,9,19,10,7,16,17)(6,13,14,18,12,11,20,15)(23,31)(27,35)(32,33)(34,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=egh,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228996);