#405965

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1276 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,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, 313161);