#238637

order	512
nilpotency class	3
isoclinism	isoclinic class of #233789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37606	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145833);