#297912

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
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×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1047 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)(5,6,10,13)(7,14,16,19)(8,12,9,18)(11,17,20,15)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,32,36,33);(1,2,4,3)(23,32)(24,25)(27,35)(28,29)(31,36)(33,34);(5,7,8,17,10,16,9,15)(6,11,18,19,13,20,12,14)(21,23,25,32,26,34,24,33)(22,27,29,35,30,36,28,31);(1,3,4,2)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,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,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, 205108);