#251780

order	512
nilpotency class	3
isoclinism	isoclinic class of #251761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18300	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 64

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

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

Polycyclic presentation:

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

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, 158976);