#298163

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1049 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)(7,9)(8,15)(10,14)(11,17)(12,18)(13,19)(16,20)(21,22,27,31)(23,28,32,35)(24,34,33,36)(25,30,26,29);(1,2,4,3)(5,7,9,6)(8,13,16,11)(10,14,18,12)(15,17,20,19)(21,23)(22,28)(24,25)(26,33)(27,32)(29,36)(30,34)(31,35);(5,8,10,17,9,16,18,19)(6,11,12,15,7,13,14,20)(21,24,25,23,27,33,26,32)(22,28,29,34,31,35,30,36);(1,3,4,2)(6,7)(11,13)(12,14)(15,20)(23,32)(24,33)(28,35)(34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=ehi,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=di

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, 205359);