#395807

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2192 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)(9,10)(11,14)(12,13)(15,16)(17,18,22,24,20,26,21,25)(19,27,30,32,28,23,29,31);(5,6)(7,8)(10,13)(14,16)(17,19,20,28)(18,23,26,27)(21,29,22,30)(24,31,25,32);(2,4)(6,8)(9,11,12,15)(10,14,13,16)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(18,25)(21,22)(23,32)(24,26)(27,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303003);