#6376276

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283472);