#312079

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,5)(3,7,8,6)(9,10,14,16,13,11,12,15)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(2,5)(6,7)(9,11,13,10)(12,15,14,16)(17,19)(18,22)(20,25)(21,27)(23,29)(24,31)(26,28)(30,32);(9,10,13,11)(12,16,14,15)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(1,3,4,8)(2,6,5,7)(9,11,13,10)(12,15,14,16)

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=h,d^2=ef,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219275);