#10690⋊C₂ #244496

order	512
nilpotency class	3
isoclinism	isoclinic class of #10690⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

unambiguous semidirect product

#10690⋊C₂

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#401
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1014 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=e,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284397, #72284424, #72284460, …
maximal normal subgroup of	#72284397, #72284424, #72284460, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151692);