#282048

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
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	#24150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

881 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)(13,14,17,15,18,21,19,22)(16,24,28,25,27,23,26,20);(2,4)(5,6)(7,9)(8,11)(10,12)(13,15,17,21,18,22,19,14)(16,25,28,23,27,20,26,24);(6,9)(7,12)(10,11)(13,16,18,27)(14,20,21,25)(15,23,22,24)(17,26,19,28);(2,4)(13,17,18,19)(14,15,21,22)(16,26,27,28)(20,23,25,24)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189244);