#298201

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1023 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

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, 205397);