#391902

order	512
nilpotency class	3
isoclinism	isoclinic class of #391902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14663
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

811 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299098);