#407934

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1039 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=ei,d^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315130);