#244249

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

786 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=ef,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151445);