#239047

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 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)(5,6,9,7)(8,10,12,11)(13,14,19,24)(15,16,26,28)(17,22,18,23)(20,21,27,25);(5,7,9,6)(8,11,12,10)(13,15,19,26)(14,20,24,27)(16,23,28,22)(17,21,18,25);(2,4)(6,10)(7,11)(8,12)(13,16,17,27,19,28,18,20)(14,21,23,15,24,25,22,26);(13,17,19,18)(14,22,24,23)(15,21,26,25)(16,27,28,20)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146243);