#6373539

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2166 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)(13,14,20,24)(15,25,16,28)(17,22,21,27)(18,23,19,26);(6,9)(10,12)(13,15,19,22,20,16,18,27)(14,21,23,25,24,17,26,28);(5,7,8,11)(6,10,9,12)(13,16,19,27,20,15,18,22)(14,17,23,28,24,21,26,25);(1,3,4,2)(13,17)(14,22)(15,26)(16,23)(18,25)(19,28)(20,21)(24,27);(13,18,20,19)(14,23,24,26)(15,27,16,22)(17,28,21,25)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280735);