#333269

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19698	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

631 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240465);