#298270

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	#39747	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,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,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, 205466);