#249764

order	512
nilpotency class	3
isoclinism	isoclinic class of #249759
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
prime radical	#60817	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1077 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1077 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)(3,10)(4,6)(5,9)(7,11)(8,14)(12,16)(13,15)(18,22)(20,27)(23,31)(24,28)(26,29)(30,32);(2,6)(4,14)(7,10)(8,9)(11,13)(15,16)(19,25)(20,27)(22,28)(23,32);(1,2,5,9)(3,7,12,15)(4,8,14,6)(10,13,16,11)(17,18)(19,23)(20,28)(21,24)(22,27)(25,32)(26,30)(29,31);(1,3)(2,7)(4,13)(5,12)(6,10)(8,16)(9,15)(11,14)(17,19,21,25)(18,23,24,32)(20,29,27,26)(22,30,28,31)

Polycyclic presentation:

2^9|c^2=h,d^2=i,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,d^b=di,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156960);