#239890

order	512
nilpotency class	3
isoclinism	isoclinic class of #233958
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1010 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)(5,6,9,7)(8,10,12,11)(13,14,18,22)(15,19,24,28)(16,21,17,20)(23,26,25,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,18,24)(14,19,22,28)(16,25,17,23)(20,26,21,27);(1,3,4,2)(6,10)(7,11)(8,12)(14,20)(16,17)(19,26)(21,22)(23,25)(27,28);(13,16,18,17)(14,21,22,20)(15,23,24,25)(19,26,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147086);