#413437

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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1022 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)(7,12)(8,10)(9,11)(13,14)(15,16)(17,21)(18,28)(19,22)(20,23)(24,26)(25,27);(2,4)(5,7,9,8)(6,10,11,12)(13,15)(14,16)(17,20)(18,24)(19,25)(21,23)(22,27)(26,28);(2,4)(5,8,9,7)(6,10,11,12)(13,14)(15,23)(16,20)(17,28)(18,21)(19,22)(24,27)(25,26);(13,16,17,26,19,27,18,23)(14,20,28,25,22,24,21,15)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,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, 320633);