#412952

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
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

1004 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 320148);