#318511

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 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:

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

Polycyclic presentation:

2^9|b^2=eghi,c^2=i,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,f^c=fi

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, 225707);