#277226

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
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

686 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)(5,6,10,11,9,7,8,12)(13,14,16,20,18,22,17,21)(15,19,23,28,25,26,24,27);(1,2,4,3)(13,15,18,25)(14,19,22,26)(16,23,17,24)(20,28,21,27);(5,7,9,6)(8,12,10,11)(14,20)(15,23)(16,17)(19,26)(21,22)(24,25);(1,3,4,2)(5,8,9,10)(6,11,7,12)(13,16,18,17)(14,21,22,20)(15,23,25,24)(19,27,26,28)

Polycyclic presentation:

2^9|a^2=fh,b^2=ei,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184422);