#477159

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1207 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)(5,6,9,8)(7,10,11,12)(13,14)(15,24)(16,19)(17,21)(18,22)(20,26)(23,28)(25,27);(1,3,4,2)(6,8)(10,12)(13,14,17,21)(15,25,16,20)(18,23,28,22)(19,27,24,26);(2,3)(5,7,9,11)(6,10,8,12)(13,15)(14,19)(16,17)(18,27)(20,22)(21,24)(23,25)(26,28);(2,3)(5,8,9,6)(7,10,11,12)(13,16,17,15)(14,20,21,25)(18,27,28,26)(19,23,24,22)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384355);