#333822

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241018);