#267368

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#32904	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1059 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=hi,d^2=i,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, 174564);