#197971

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1444 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105167);