#249758

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1052 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,12)(4,14)(5,8)(6,11)(7,10)(9,16)(13,15)(17,18,22,20)(19,25,27,32)(21,29,30,24)(23,31,26,28);(1,3,7,13)(2,5,10,4)(6,14,16,8)(9,15,11,12)(21,30)(24,29)(25,32)(28,31);(1,4,7,5)(2,8,10,14)(3,9,13,11)(6,15,16,12)(17,19,22,27)(18,23,20,26)(21,31,30,28)(24,32,29,25);(1,5,7,4)(2,3,10,13)(6,15,16,12)(8,9,14,11)(17,20,22,18)(19,26,27,23)(21,24,30,29)(25,31,32,28)

Polycyclic presentation:

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

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, 156954);