#380053

order	512
nilpotency class	3
isoclinism	isoclinic class of #29730
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1598 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)(10,11)(14,15)(17,18,22,26,20,23,21,25)(19,27,30,24,28,32,29,31);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(24,31)(25,26)(27,32);(2,4)(6,8)(9,11,13,10)(12,14,16,15)(17,19,20,28)(18,24,23,31)(21,30,22,29)(25,32,26,27);(9,12,13,16)(10,14,11,15)(19,28)(24,31)(27,32)(29,30)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287249);