#459343

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1178 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366539);