#234020

order	512
nilpotency class	3
isoclinism	isoclinic class of #234006
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1456 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,e^b=eh,f^c=fi

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, 141216);