#6370617

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2172 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277813);