#312053

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=h,d^2=efh,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, 219249);