#272950

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180146);