#339837

order	512
nilpotency class	3
isoclinism	isoclinic class of #339775
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

528 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247033);