#6370503

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf,e^c=ei,e^d=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277699);