#277077

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

693 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)(5,6,9,10,8,12,7,11)(13,14)(15,19)(16,21)(17,20)(18,22)(23,28)(24,26)(25,27);(1,2,4,3)(5,7,8,9)(6,10,12,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,26)(22,28);(6,11)(7,9)(10,12)(14,20)(15,23)(16,18)(19,26)(24,25)(27,28);(1,3,4,2)(6,12)(10,11)(13,16,17,18)(14,21,20,22)(15,24,25,23)(19,26,27,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184273);