#413534

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1184 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320730);